This Maple file accompanies the paper "Asymptotics of minimal deterministic finite automata recognizing a finite binary language"
by Andrew Elvey Price, Wenjie Fang and Michael Wallner.
#############################################################################################################################
Abstract:
We show that the number of minimal discrete finite automata with n transient states recognizing
a finite binary language grows asymptotically for n to infinity like
Theta( n! \134, 4^n e^( 3a1 n^(1/3) ) n^(7/8) ),
where a1 is approximately -2.338 and the largest root of the Airy function.
For this purpose, we derive a new two parameter recurrence relation which yields an algorithm of
quadratic arithmetic complexity for computing the number of such automata up to a given size.
Using this result, we prove by induction asymptotically tight bounds that are sufficiently accurate
for large n to determine the asymptotic form using adapted Netwon polygons.
#############################################################################################################################
In this worksheet
*) we compute the initial terms of the recurrences
*) compute the Newton polygons in the proof so Lemmas 6 and 7
restart;
with(plots):
Recurrences
In some computations at the end we use the gfun-package.
It is just needed to guess simple functional equations for generating functions and manipulate their coefficient.
This is not needed in the remainder of this worksheet.
For the latest version see Salvy's homepage http://perso.ens-lyon.fr/bruno.salvy/software/the-gfun-package/.
For these computations gfun verison 3.76 was used.
#libname := "<path>\134\134gfun.mla", libname; #set the path to gfun.mla
libname := "D:\134\134lib\134\134gfun.mla", "D:\134\134lib", "/home/michaelwallner/lib/gfun.mla", "home/michaelwallner/lib", libname;
with(gfun): gfun:-version;
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShsaWJuYW1lR0YoNidRMEQ6XGxpYlxnZnVuLm1sYUYoUSdEOlxsaWJGKFFCL2hvbWUvbWljaGFlbHdhbGxuZXIvbGliL2dmdW4ubWxhRihROGhvbWUvbWljaGFlbHdhbGxuZXIvbGliRihROS91c3IvbG9jYWwvTWFwbGUyMDE3L2xpYkYoNydGL0YwRjFGMkYz
JCIkdyQhIiM=
Maximal number of computed terms
NN := 100;
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNOTkdGKCIkKyI3I0Yu
Minimal binary DFAs or B-paths
Compute the minimal binary DFAs of size up to NN
(Recall: size is the number of transient states, i.e., not counting the unique recurrent state)
for n from 0 to NN do
for m from 0 to NN do
bb[n,m] := 0:
end:
end:
#initial conditions
bb[0,0] := 1:
for n from 1 to NN do
bb[n,0] := 1:
bb[n,1] := bb[n,0] + 2*bb[n-1,1]:
end:
for n from 1 to NN do
for m from 2 to n do
bb[n,m] := 2*bb[n,m-1]+(m+1)*bb[n-1,m]-m*bb[n-2,m-1];
end do:
end do:
print the array
for m from 7 to 0 by -1 do
for n from 0 to 10 do
printf("%10.0f ",bb[n,m]);
end;
printf("\134n");
end;
0 0 0 0 0 0 0 15824880 306252120 4211946600 49545344760
0 0 0 0 0 0 487560 7912440 91533000 908658360 8245251480
0 0 0 0 0 18480 243780 2305200 18804300 140879280 998734380
0 0 0 0 900 9240 68700 444360 2658300 15137640 83375100
0 0 0 60 450 2490 12150 55410 242550 1033890 4328550
0 0 6 30 114 390 1266 3990 12354 37830 115026
0 1 3 7 15 31 63 127 255 511 1023
1 1 1 1 1 1 1 1 1 1 1
JSFH
check
According to [Domaratzki, Kisman, Shallit 2002] page 16 this should be equal to f'_2(n)
1, 1, 6, 60, 900, 18480, 487560
seq(bb[n,n],n=0..min(NN,12));
Ni8iIiJGIyIiJyIjZyIkKyoiJiFbPSInZ3ZbIikhKVsjZSIiKlNVXTcnIixTWW0+dyMiLlMtKFszRDkiL2dkb2N0JEgpIjFTJzMrKCpcN1Em
Minimal binary DFAs or B-paths divided by 2^(n-1)
Compute the relaxed binary trees of size up to NN
for n from 0 to NN do
for m from 0 to NN do
cc[n,m] := 0:
end:
end:
#initial conditions
cc[0,0] := 2:
for n from 1 to NN do
cc[n,0] := 1:
cc[n,1] := cc[n,0] + 2*cc[n-1,1]:
end:
for n from 1 to NN do
for m from 2 to n do
cc[n,m] := cc[n,m-1]+(m+1)*cc[n-1,m]-m/2*cc[n-2,m-1];
end do:
end do:
print the array
for m from 7 to 0 by -1 do
for n from 0 to 10 do
printf("%10.0f ",cc[n,m]*2^(n-1));
end;
printf("\134n");
end;
0 0 0 0 0 0 0 15824880 612504240 16847786400 396362758080
0 0 0 0 0 0 487560 15824880 366132000 7269266880 131924023680
0 0 0 0 0 18480 487560 9220800 150434400 2254068480 31959500160
0 0 0 0 900 18480 274800 3554880 42532800 484404480 5336006400
0 0 0 60 900 9960 97200 886560 7761600 66168960 554054400
0 0 6 60 456 3120 20256 127680 790656 4842240 29446656
0 1 6 28 120 496 2016 8128 32640 130816 523776
1 1 2 4 8 16 32 64 128 256 512
check
According to [Domaratzki, Kisman, Shallit 2002] page 16 this should be equal to f'_2(n)
1, 1, 6, 60, 900, 18480, 487560
seq(cc[n,n]*2^(n-1),n=0..min(NN,12));
Ni8iIiJGIyIiJyIjZyIkKyoiJiFbPSInZ3ZbIikhKVsjZSIiKlNVXTcnIixTWW0+dyMiLlMtKFszRDkiL2dkb2N0JEgpIjFTJzMrKCpcN1Em
The array without rescaling
(*
for m from 6 to 0 by -1 do
for n from 0 to 8 do
printf("%10.2f ",cc[n,m]);
end;
printf("\134n");
end;
*)
Weighted minimal binary DFAs
Maximal number of meanders computed
NN:=100:
compute the meanders of size up to NN
for n from -1 to NN do
for m from -1 to NN do
ee[n,m] := 0:
end:
end:
ee[0,0] := 2:
ee[1,1] := 1:
for n from 2 to NN do
ee[n,0] := ee[n-1,1];
for m from 1 to n do
ee[n,m] := (n-m+2)/(n+m)*ee[n-1,m-1]+ee[n-1,m+1]-(n-m)/(n+m)/(n+m-2)*ee[n-3,m-1];
end:
#special case for bb[n,1]
ee[n,n-2] := 2/(n-1)*ee[n-1,n-3]+ee[n-1,n-1];
end:
print the array
for m from 6 to 0 by -1 do
for n from 0 to 11 do
printf("%8.0f ",ee[n,m]*factorial(floor((n+m)/2))*2^((n+m)/2-1));
end;
printf("\134n");
end;
0 0 0 0 0 0 32 0 8128 0 790656 0
0 0 0 0 0 16 0 2016 0 127680 0 7761600
0 0 0 0 8 0 496 0 20256 0 886560 0
0 0 0 4 0 120 0 3120 0 97200 0 3554880
0 0 2 0 28 0 456 0 9960 0 274800 0
0 1 0 6 0 60 0 900 0 18480 0 487560
1 0 1 0 6 0 60 0 900 0 18480 0
check
seq(factorial(n/2)*ee[n,0]*2^((n)/2-1),n=0..min(NN,22),2);
seq(bb[n,n],n=0..min(NN,11));
Ni4iIiJGIyIiJyIjZyIkKyoiJiFbPSInZ3ZbIikhKVsjZSIiKlNVXTcnIixTWW0+dyMiLlMtKFszRDkiL2dkb2N0JEgp
Ni4iIiJGIyIiJyIjZyIkKyoiJiFbPSInZ3ZbIikhKVsjZSIiKlNVXTcnIixTWW0+dyMiLlMtKFszRDkiL2dkb2N0JEgp
We guess a few simple formulas for the numbers ending close to the upper end.
There is just 1 path ending on the very top
seq(ee[n,n]*factorial(n),n=2..20);
NjUiIiJGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiM=
n:='n':
seq(ee[n,n-2]*factorial(n-1),n=2..20);
listtorec([%],a(n));
rsolve(op(1,%),a(n));
simplify(subs(n=n-2,%));
NjUiIiIiIiQiIigiIzoiI0oiI2oiJEYiIiRiIyIkNiYiJUI1IiVaPyIlJjQlIiUiPikiJiRROyImbkYkIiZOYiciJ3I1OCInVkBFIicoR0Mm
NyQ8JSwoKiYiIiMiIiItSSJhRzYiNiNJIm5HRipGJ0YnKiYiIiRGJy1GKTYjLCZGLEYnRidGJ0YnISIiLUYpNiMsJkYsRidGJkYnRicvLUYpNiMiIiFGJy8tRik2I0YnRi5JJG9nZkdGKg==
LCYqJiIiIyIiIilGJEkibkc2IkYlRiVGJSEiIg==
LCYpIiIjLCZJIm5HNiIiIiJGKCEiIkYoRihGKQ==
n:='n':
seq(ee[n,n-4]*factorial(n-2),n=4..20);
listtorec([%],a(n));
rsolve(op(1,%),a(n));
map(simplify,subs(n=n-4,%)) assuming n::posint;
NjMiIiQiIzoiI2QiJCY+IiRMJyIlJio+IiV4aCImOio9IiY4diYiJ3ZTPCInKEhEJiIoTj9lIiIoJFJlWiIpYigqSDkiKTwlW0glIipiTiUqRyIiKnRzLShR
NyQ8JSwoKiYiIiciIiItSSJhRzYiNiNJIm5HRipGJ0YnKiYiIiZGJy1GKTYjLCZGLEYnRidGJ0YnISIiLUYpNiMsJkYsRiciIiNGJ0YnLy1GKTYjIiIhIiIkLy1GKTYjRiciIzpJJG9nZkdGKg==
LCYqJiIiJyIiIikiIiNJIm5HNiJGJSEiIiomIiIqRiUpIiIkRihGJUYl
LCYqJiMiIiQiIikiIiIpIiIjSSJuRzYiRichIiIqJiNGJyIiKkYnKUYlRipGJ0Yn
n:='n':
seq(ee[n,n-6]*factorial(n-3),n=6..20);
listtorec([%],a(n));
rsolve(op(1,%),a(n));
simplify(subs(n=n-6,%));
NjEiIzojIiREIyIiIyMiJVg3RiYjIiV2Z0YmIyImMHgjRiYjIid2NzdGJiMiJ1hwXkYmIyIodlU7I0YmIyIoMCZbKilGJiMiKXY/bk9GJiMiKlgsTFwiRiYjIip2OUUwJ0YmIyIrMCQqKltXI0YmIyIrdik9NyYpKkYmIyIsWEwnKT4nUkYm
NyQ8JiwqKiYiI0MiIiItSSJhRzYiNiNJIm5HRipGJyEiIiomIiNFRictRik2IywmRixGJ0YnRidGJ0YnKiYiIipGJy1GKTYjLCZGLEYnIiIjRidGJ0YtLUYpNiMsJkYsRiciIiRGJ0YnLy1GKTYjIiIhIiM6Ly1GKTYjRicjIiREI0Y4Ly1GKTYjRjgjIiVYN0Y4SSRvZ2ZHRio=
LCgqJiIjdiIiIikiIiVJIm5HNiJGJUYlKiYjIiROIiIiI0YlKSIiJEYoRiUhIiIqJiMiIzpGLUYlKUYtRihGJUYl
LCgqJiMiI3YiJSc0JSIiIikiIiVJIm5HNiJGJ0YnKiYjIiImIiNhRicpIiIkRipGJyEiIiomIyIjOiIkRyJGJykiIiNGKkYnRic=
(*
for m from 6 to 0 by -1 do
for n from 0 to 11 do
printf("%8.2f ",ee[n,m]);
end;
printf("\134n");
end;
*)
Check lower and upper bounds
NN;
IiQrIg==
(*
m:=10;
seq(mysign(-ee[n,m] + (n-m+2)/(n+m)*ee[n-1,m-1] + (n-m-1)/(n-m)*ee[n-1,m+1]),n=max(3,m+1)..30);
seq(mysign(-1/(n-m)*ee[n-1,m+1] + 1/(n+m)*ee[n-2,m] + (n-m-3)/(n-m)/(n-m-2)*ee[n-2,m+2]),n=max(m+3,5)..30);
*)
This should always be negative
for n from -1 to NN do
for m from -1 to NN do
eelow[n,m] := 0:
end:
end:
for n from 3 to NN do
for m from 1 to n-3 do
eelow[n,m] := -ee[n,m] + (n-m+2)/(n+m)*ee[n-1,m-1] + (n-m-1)/(n-m)*ee[n-1,m+1] + (n-m-3)/(n-m-2)*(1/(n-m)*ee[n-2,m+2] + 1/(n+m)*ee[n-3,m+1]);
end:
end:
mysign := proc(i) if i=0 then return 0 else return sign(i) end end:
looks good
for m from 16 to 0 by -1 do
for n from 5 to 20 do
printf("%3.0f ",mysign(eelow[n,m]*factorial(floor((n+m)/2))));
end;
printf("\134n");
end;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1
0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0
0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 -1
0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 -1 0
0 0 0 0 0 0 0 0 0 -1 0 -1 0 -1 0 -1
0 0 0 0 0 0 0 0 -1 0 -1 0 -1 0 -1 0
0 0 0 0 0 0 0 -1 0 -1 0 -1 0 -1 0 -1
0 0 0 0 0 0 -1 0 -1 0 -1 0 -1 0 -1 0
0 0 0 0 0 -1 0 -1 0 -1 0 -1 0 -1 0 -1
0 0 0 0 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0
0 0 0 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0 -1
0 0 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0
0 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0 -1
-1 0 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
n0:=51:
seq(mysign(eelow[n0,m]),m=n0 mod 2..n0,2);
NjwhIiJGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjIiIhRiQ=
This should always be non-negative
for n from -1 to NN do
for m from -1 to NN do
eeup[n,m] := 0:
end:
end:
for n from 3 to NN do
for m from 1 to n-3 do
eeup[n,m] := -ee[n,m] + (n-m+2)/(n+m)*ee[n-1,m-1] + (n-m-1)/(n-m)*ee[n-1,m+1] + 1/(n-m)*ee[n-2,m+2] + 1/(n+m)*ee[n-3,m+1];
end:
end:
n0:=51:
seq(mysign(eeup[n0,m]),m=n0 mod 2..n0,2);
NjwiIiJGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGIyIiIUYkRiQ=
looks good
for m from 16 to 0 by -1 do
for n from 5 to 20 do
printf("%3.0f ",mysign(eeup[n,m]*factorial(floor((n+m)/2))));
end;
printf("\134n");
end;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0
0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1
0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0
0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1
0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0
0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1
0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0
0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1
0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
m := 'm':
n := 'n':
Newton polygons
Programs
look at possible degrees of each term in n and m
x-axis m degree
y-axis n degree
mynewt := proc(F,m,n)
local ss,newt,tt,mdeg,ndeg;
ss := 0;
newt := {}:
for tt in expand(collect(F,[m,n],simplify)) do
mdeg := simplify(m*diff(tt,m)/tt);
ndeg := simplify(n*diff(tt,n)/tt);
newt := {op(newt), [mdeg,ndeg]};
end:
return newt;
end:
Get the term of order Theta(m^a*n^b) from the Newton polygon computed using mynewt
getel := proc(tmp,a,b)
local tt,mdeg,ndeg,ret:
ret := 0:
for tt in expand(tmp) do
mdeg := m*diff(tt,m)/tt;
ndeg := n*diff(tt,n)/tt;
if mdeg=a and ndeg=b then ret := ret + tt end;
end:
return ret;
end:
Get the maximal power of n^b for each m^a for a=0..M
getMaxNewt := proc(M::posint, newt)
local i, el, mnmax;
for i from 0 to M do mnmax[i]:=-infinity end:
for el in newt do
if el[1]<=M then
if mnmax[el[1]]<el[2] then mnmax[el[1]]:=el[2] end:
end;
end;
return mnmax;
end:
Compute the slopes and corners of the convex hull
maxslope := proc(ll,M,i)
local j,sl,tmp,sj;
sl := ll[i+1]-ll[i]: #initial slope between first 2 points
#is this the one of the hull? find max starting from 0
sj:=M;
for j from i+1 to M do
tmp := (ll[j]-ll[i])/(j-i);
if tmp >= sl then sl:=tmp;sj:=j; end;
end:
return sl,sj;
end:
Compute the slopes and corners of the convex hull
getslopes := proc(ll,M)
local sl,sj,li,ls;
sj:=0;
li:=sj:
ls:=0:
#go on and find other slopes of convex hull
while sj<M do
sl,sj := maxslope(ll,M,sj):
#save it
li:=li,sj;
ls:=ls,sl;
end:
li:=[li];
ls:=[seq(ls[i],i=2..nops(li))];
return ls,li;
end:
Shorthands
The largest root of the Airy funciton Ai(z)
A1 := AiryAiZeros(1);
evalf(%);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNBMUdGKC1JLEFpcnlBaVplcm9zR0YlNiMiIiI3I0Yu
JCErNXU1UUIhIio=
This is the constant c of Equation (6):
s(n) = 2 + c*n^(-2/3) + ...
(Also called sigma_2 in the ansatz used in the proof of Lemma 4.2)
csubs := isolate((1/2)*2^(1/3)*(c) = a1,c);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjc3Vic0dGKC9JImNHRigqJkkjYTFHRigiIiIpIiIjI0Y0IiIkRjI3I0Yu
Labeling options to produce nice plots.
myoptionsLo := labels=["i", "j"],symbolsize=25, symbol=diamond,axesfont = ["HELVETICA", "ROMAN", 15],labelfont = ["HELVETICA", 18]:
myoptionsUp := labels=["i", "j"],symbolsize=20, symbol=diamond,axesfont = ["HELVETICA", "ROMAN", 15],labelfont = ["HELVETICA", 18]:
We introduce the following shorthands for the Airy function and its derivative
kaplam := AiryAi(AiryAiZeros(1)+2^(1/3)*m/n^(1/3))=kappa, AiryAi(1, AiryAiZeros(1)+2^(1/3)*m/n^(1/3))=lambda;
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdrYXBsYW1HRig2JC8tSSdBaXJ5QWlHRiU2IywmLUksQWlyeUFpWmVyb3NHRiU2IyIiIkY3KigpIiIjI0Y3IiIkRjdJIm1HRihGNylJIm5HRihGOyEiIkY3SSZrYXBwYUdGKC8tRjE2JEY3RjNJJ2xhbWJkYUdGKDckRi9GQg==
Expansions
We expand the Airy function around a1+2^(1/3)*m/n^(1/3) up to chosen order ordAi
FFy := AiryAi(AiryAiZeros(1)+2^(1/3)*m/n^(1/3)+y);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRGRnlHRigtSSdBaXJ5QWlHRiU2IywoLUksQWlyeUFpWmVyb3NHRiU2IyIiIkY1KigpIiIjI0Y1IiIkRjVJIm1HRihGNSlJIm5HRihGOSEiIkY1SSJ5R0YoRjU3I0Yu
We use two different expansion orders for the upper and lower bound
(to speed up the computations, and to produce the best pictures)
Here, we start with the lower bound , which needs less terms
ordAiLo := 13;
FFyserLo := map(expand,series(FFy,y,ordAiLo)):
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShvcmRBaUxvR0YoIiM4NyNGLg==
For y = x-(a1+2^(1/3)*m/n^(1/3)) we have that FF(x)=Ai(x),
i.e. an expansion of the Airy function
FFxserLo := subs(y=x-(AiryAiZeros(1)+2^(1/3)*m/n^(1/3)),FFyserLo):
Replace the appearing Airy functions by our shorthands kappa and lambda
indets(FFxserLo);
FFxserLoKL := subs(kaplam,AiryAiZeros(1)=a1,FFxserLo):
indets(%);
PCtJIm1HNiJJIm5HRiRJInhHRiQqJClGJSMiIiYiIiQhIiIqJClGJSMiIiVGK0YsKiQpRiUjIiIjRitGLCokKUYlIyIiIkYrRiwtSSdBaXJ5QWlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IywmLUksQWlyeUFpWmVyb3NHRjs2I0Y4RjgqKClGNEY3RjhGI0Y4RjZGLEY4LUY6NiRGOEY/
PCxJI2ExRzYiSSZrYXBwYUdGJEknbGFtYmRhR0YkSSJtR0YkSSJuR0YkSSJ4R0YkKiQpRigjIiImIiIkISIiKiQpRigjIiIlRi5GLyokKUYoIyIiI0YuRi8qJClGKCMiIiJGLkYv
Then, we use the generic ansatz for factor of FFy, i.e. the Airy function
facAiryLo := (1+(add(q[i]*m^i,i=0..2))/(n));
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpmYWNBaXJ5TG9HRigsJiIiIkYvKiYsKComKUkibUdGKCIiI0YvJkkicUdGKDYjRjVGL0YvKiZGNEYvJkY3NiNGL0YvRi8mRjc2IyIiIUYvRi9JIm5HRighIiJGLzcjRi4=
This is our ansatz
(only the substitution is influenced by the parameters, i.e. not the ms and ns that are already in FFxser;
that is what we want, as all of them should be expanded at a1+2^(1/3)*m/n(1/3))
Note that due to the replacement with kappa and lambda, the expansions are fixed already at this point and the ms and ns in the arguments of Ai and Ai' are not influenced.
Note that we substitute now a1+2^(1/3)*(m+1)/n^(1/3), i.e. m+1 instead of m around which we expanded above.
XFL := (n0,m0) -> subs(n=n0,m=m0,facAiryLo)*subs(x=a1+2^(1/3)*(m0+1)/n0^(1/3),FFxserLoKL);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRYRkxHRihmKjYkSSNuMEdGKEkjbTBHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKComLUklc3Vic0dGJjYlL0kibkdGKEYwL0kibUdGKEYxSSpmYWNBaXJ5TG9HRigiIiItRjc2JC9JInhHRigsJkkjYTFHRihGPiooKSIiIyNGPiIiJEY+LCZGMUY+Rj5GPkY+KUYwRkghIiJGPkkrRkZ4c2VyTG9LTEdGKEY+RihGKEYoNyNGLg==
Then, we do the same for the upper bound , with a few more terms
ordAiUp := 19;
FFyserUp := map(expand,series(FFy,y,ordAiUp)):
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShvcmRBaVVwR0YoIiM+NyNGLg==
For y = x-(a1+2^(1/3)*m/n^(1/3)) we have that FF(x)=Ai(x),
i.e. an expansion of the Airy function
FFxserUp := subs(y=x-(AiryAiZeros(1)+2^(1/3)*m/n^(1/3)),FFyserUp):
Replace the appearing Airy functions by our shorthands kappa and lambda
indets(FFxserUp);
FFxserUpKL := subs(kaplam,AiryAiZeros(1)=a1,FFxserUp):
indets(%);
PC1JIm1HNiJJIm5HRiRJInhHRiQqJClGJSMiIikiIiQhIiIqJClGJSMiIihGK0YsKiQpRiUjIiImRitGLCokKUYlIyIiJUYrRiwqJClGJSMiIiNGK0YsKiQpRiUjIiIiRitGLC1JJ0FpcnlBaUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCYtSSxBaXJ5QWlaZXJvc0dGQzYjRkBGQCooKUY8Rj9GQEYjRkBGPkYsRkAtRkI2JEZARkc=
PC5JI2ExRzYiSSZrYXBwYUdGJEknbGFtYmRhR0YkSSJtR0YkSSJuR0YkSSJ4R0YkKiQpRigjIiIpIiIkISIiKiQpRigjIiIoRi5GLyokKUYoIyIiJkYuRi8qJClGKCMiIiVGLkYvKiQpRigjIiIjRi5GLyokKUYoIyIiIkYuRi8=
In the upper bound, we use a more generic factor of FFy, there are the additional terms p[i] which are divided by n^2
facAiryUp := (1+(add(p[i]*m^i,i=0..4))/(n^2)+(add(q[i]*m^i,i=0..2))/(n));
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpmYWNBaXJ5VXBHRigsKCIiIkYvKiYsLComKUkibUdGKCIiJUYvJkkicEdGKDYjRjVGL0YvKiYpRjQiIiRGLyZGNzYjRjtGL0YvKiYpRjQiIiNGLyZGNzYjRkBGL0YvKiZGNEYvJkY3NiNGL0YvRi8mRjc2IyIiIUYvRi8pSSJuR0YoRkAhIiJGLyomLCgqJkY/Ri8mSSJxR0YoRkJGL0YvKiZGNEYvJkZQRkVGL0YvJkZQRkdGL0YvRkpGS0YvNyNGLg==
This is our ansatz for the upper bound (comparable to XFL)
(only the substitution is influenced by the parameters, i.e. not the ms and ns that are already in FFxser;
that is what we want, as all of them should be expanded at a1+2^(1/3)*m/n(1/3))
XFU := (n0,m0) -> subs(n=n0,m=m0,facAiryUp)*subs(x=a1+2^(1/3)*(m0+1)/n0^(1/3),FFxserUpKL);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRYRlVHRihmKjYkSSNuMEdGKEkjbTBHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKComLUklc3Vic0dGJjYlL0kibkdGKEYwL0kibUdGKEYxSSpmYWNBaXJ5VXBHRigiIiItRjc2JC9JInhHRigsJkkjYTFHRihGPiooKSIiIyNGPiIiJEY+LCZGMUY+Rj5GPkY+KUYwRkghIiJGPkkrRkZ4c2VyVXBLTEdGKEY+RihGKEYoNyNGLg==
Finally, the ansatz for the quotient of h(n)/h(n-1).
Note that pterm is a mnemonic for "polynomial term", as this value influences the polynomial term n^{alpha};
The other values have similar interpretations:
a .... exponantial growth, i.e. a^n
b .... b will be zero
c .... stretched exponential
d .... technical choice, to simplify the proofs
SF := n -> a+b/n^(1/3)+c/n^(2/3)+pterm/n + d/n^(7/6);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTRkdGKGYqNiNJIm5HRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwsSSJhR0YoIiIiKiZJImJHRihGNilGMCNGNiIiJCEiIkY2KiZJImNHRihGNilGMCMiIiNGO0Y8RjYqJkkmcHRlcm1HRihGNkYwRjxGNiomSSJkR0YoRjYpRjAjIiIoIiInRjxGNkYoRihGKDcjRi4=
Minimal DFAs recoginizing a finite binary language - Lower bound
Xtilde
This is Y tilde
Xansatz := (n,m) -> (1-2*(m)^2/(3*n)+3*m/(8*n))* AiryAi(a1+2^(1/3)*(m+1)/n^(1/3));
Zio2JEkibkc2IkkibUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYsKCIiIkYsKigjIiIjIiIkRiwpRiZGL0YsRiQhIiJGMiooI0YwIiIpRixGJkYsRiRGMkYsRiwtSSdBaXJ5QWlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywmSSNhMUdGJUYsKigpRi8jRixGMEYsLCZGJkYsRixGLEYsKUYkRkBGMkYsRixGJUYlRiU=
Sansatz := n -> 2 + c*n^(-2/3) + pterm/n - 1/(n^(7/6));
Zio2I0kibkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCoiIiMiIiIqJkkiY0dGJUYrKUYkI0YqIiIkISIiRisqJkkmcHRlcm1HRiVGK0YkRjFGKyokKUYkIyIiKCIiJ0YxRjFGJUYlRiU=
lower bound
posansatz := -XX(n,m)*SS(n)*SS(n-1)*SS(n-2)
+ (n-m+2)/(n+m)*XX(n-1,m-1)*SS(n-1)*SS(n-2)
+ (n-m-1)/(n-m)*XX(n-1,m+1)*SS(n-1)*SS(n-2)
+ (n-m-3)/(n-m-2)*( 1/(n-m)*XX(n-2,m+2)*SS(n-2) + 1/(n+m)*XX(n-3,m+1) )
;
LCoqKi1JI1hYRzYiNiRJIm5HRiZJIm1HRiYiIiItSSNTU0dGJjYjRihGKi1GLDYjLCZGKEYqRiohIiJGKi1GLDYjLCZGKEYqIiIjRjFGKkYxKiwsKEYoRipGKUYxRjVGKkYqLCZGKEYqRilGKkYxLUYlNiRGMCwmRilGKkYqRjFGKkYuRipGMkYqRioqLCwoRihGKkYpRjFGKkYxRiosJkYoRipGKUYxRjEtRiU2JEYwLCZGKUYqRipGKkYqRi5GKkYyRipGKiooLChGKEYqRilGMSIiJEYxRiosKEYoRipGKUYxRjVGMUYxLCYqKEY+RjEtRiU2JEY0LCZGKUYqRjVGKkYqRjJGKkYqKiZGOEYxLUYlNiQsJkYoRipGREYxRkFGKkYqRipGKg==
posXS := map(simplify,subs(XX=Xansatz,SS=Sansatz,posansatz)):
For a large n this function of m seems to be positive
Digits:=20:
e1 := subs(csubs,pterm=29/12,a1=A1,posXS):
N := 100000;
M := floor(N^(1/2));
P1 := plot([seq([mm,(subs(n=N,m=mm,e1))],mm=0..M)]):display(P1);
IicrKzU=
IiQ7JA==
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Prove it
We start with the ansatz of Ytilde in Lemma 5.2.
Recall the general ansatz
facAiryLo*Airy(a1+2^(1/3)*(m+1)/n^(1/3));
SF(n);
KiYsJiIiIkYkKiYsKComKUkibUc2IiIiI0YkJkkicUdGKjYjRitGJEYkKiZGKUYkJkYtNiNGJEYkRiQmRi02IyIiIUYkRiRJIm5HRiohIiJGJEYkLUklQWlyeUdGKjYjLCZJI2ExR0YqRiQqKClGKyNGJCIiJEYkLCZGKUYkRiRGJEYkKUY1Rj5GNkYkRiQ=
LCxJImFHNiIiIiIqJkkiYkdGJEYlKUkibkdGJCNGJSIiJCEiIkYlKiZJImNHRiRGJSlGKSMiIiNGK0YsRiUqJkkmcHRlcm1HRiRGJUYpRixGJSomSSJkR0YkRiUpRikjIiIoIiInRixGJQ==
Substitute ansatz into the sequence we want to be positive for large n and all m
posF := map(expand, subs(XX=XFL,SS=SF,posansatz)):indets(%);
PDtJImFHNiJJI2ExR0YkSSJiR0YkSSJjR0YkSSJkR0YkSSZrYXBwYUdGJEknbGFtYmRhR0YkSSJtR0YkSSJuR0YkSSZwdGVybUdGJCZJInFHRiQ2IyIiISZGLzYjIiIiJkYvNiMiIiMqJClGLCMiIigiIichIiIqJClGLCMiIiYiIiRGPSokKUYsIyIiJUZCRj0qJClGLCNGN0ZCRj0qJClGLCNGNEZCRj0qJCksJkYsRjRGQkY9RkxGPSokKSwmRixGNEY3Rj1GOkY9KiQpRlJGSUY9KiQpRlJGTEY9KiQpLCZGLEY0RjRGPUY6Rj0qJClGWUZJRj0qJClGWUZMRj0=
The error terms are (to check, expand posF)
simplify(O((2^(1/3)*(m+1)/n^(1/3)-2^(1/3)*m/n^(1/3))^ordAiLo));
simplify(O((2^(1/3)*(m)/(n-1)^(1/3)-2^(1/3)*m/n^(1/3))^ordAiLo));
simplify(O((2^(1/3)*(m+2)/(n-1)^(1/3)-2^(1/3)*m/n^(1/3))^ordAiLo));
LUkiT0clKnByb3RlY3RlZEc2IywkKigiIzsiIiIpIiIjI0YpIiIkRikpSSJuRzYiIyIjOEYtISIiRik=
LUkiT0clKnByb3RlY3RlZEc2IywkKi4iIzsiIiIpIiIjI0YpIiIkRikpSSJtRzYiIiM4RikpLCYqJCksJkkibkdGMEYpRikhIiJGLEYpRikqJClGN0YsRilGOEYxRikpRjYjRjFGLUY4KUY3RjxGOEY4
LUkiT0clKnByb3RlY3RlZEc2IywkKiwiIzsiIiIpIiIjI0YpIiIkRikpLCgqJkkibUc2IkYpKSwmSSJuR0YyRilGKSEiIkYsRilGKSomKUY1RixGKUYxRilGNiomRitGKUY4RilGNiIjOEYpKUY0I0Y6Ri1GNilGNUY8RjZGNg==
remove error terms
posFd := convert(posF,polynom):
Next we fit the display size of the plots in the n (Nord) and m (Mord) direction.
(Note that everything up to ordAi is computed, but possibly not shown)
Nord := -ordAiLo/3;
Mord := floor(ordAiLo/3)+1;
myview := view=[0..Mord,Nord..0]:
IyEjOCIiJA==
IiIm
Expand again with respect to n,
these are then our unknowns
posFe := series(posFd,n=infinity,ceil(-Nord)+1):indets(%);
posFf := convert(%%,polynom):
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
The mynewt function computes the Newton polygon of posFf
newt1 := mynewt(posFf,m,n):
First Newton polygon, where no unknowns have been fixed
P1 := pointplot(newt1,myoptionsLo,color=blue):
display(P1,myview);
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Ig==
Here, we want to kill the element (0,0)
getel(posFf,0,0);
LCYqJilJImFHNiIiIiQiIiJJJmthcHBhR0YmRighIiIqKCIiI0YoKUYlRixGKEYpRihGKA==
Set a=2
posFfa := expand(simplify(subs(a=2,posFf))) assuming n::posint,m::posint:
newta := mynewt(posFfa,m,n):
All blue points have been eliminated, and only the red ones remain
P1a := pointplot(newta,myoptionsLo,color=red):
display(P1,P1a,myview);
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
b=0 is forced due to the term n^{-1/3)
getel(posFfa,0,-1/3);
LCQqKiIiJSIiIilJIm5HNiIjRiUiIiQhIiJJJmthcHBhR0YoRiVJImJHRihGJUYr
set a=2, b=0
posFfab := expand(simplify(subs(b=0,posFfa))) assuming n::posint,m::posint:
newtab := mynewt(posFfab,m,n):
Now only the green points remain.
P1ab := pointplot(newtab,myoptionsLo,color=green):
display(P1,P1a,P1ab,myview);
6--%'POINTSG6^q7$$""!!""$!#S!""7$$""!!""$!#I!""7$$""!!""$!#?!""7$$""!!""$!#5!""7$$""!!""$""!!""7$$""!!""$!1LLLLLLL[!#:7$$""!!""$!1nmmmmmmT!#:7$$""!!""$!2NLLLLLL$Q!#;7$$""!!""$!2lmmmmmm;$!#;7$$""!!""$!2NLLLLLL$G!#;7$$""!!""$!1nmmmmmmY!#:7$$""!!""$!1LLLLLLLV!#:7$$""!!""$!2lmmmmmm;#!#;7$$""!!""$!2lmmmmmmm$!#;7$$""!!""$!2LLLLLLL$=!#;7$$""!!""$!2NLLLLLLL$!#;7$$""!!""$!#X!""7$$""!!""$!2lmmmmmmm#!#;7$$""!!""$!#N!""7$$""!!""$!2NLLLLLLL#!#;7$$""!!""$!2nmmmmmm;"!#;7$$""!!""$!#D!""7$$""!!""$!2nmmmmmmm"!#;7$$""!!""$!2LLLLLLLL"!#;7$$""!!""$!#:!""7$$""!!""$!1mmmmmmmm!#;7$$""!!""$!1LLLLLLLL!#;7$$"#5!""$!#S!""7$$"#5!""$!#I!""7$$"#5!""$!#?!""7$$"#5!""$!#5!""7$$"#5!""$!1LLLLLLL[!#:7$$"#5!""$!1nmmmmmmT!#:7$$"#5!""$!2NLLLLLL$Q!#;7$$"#5!""$!2lmmmmmm;$!#;7$$"#5!""$!2NLLLLLL$G!#;7$$"#5!""$!1nmmmmmmY!#:7$$"#5!""$!1LLLLLLLV!#:7$$"#5!""$!2lmmmmmm;#!#;7$$"#5!""$!2lmmmmmmm$!#;7$$"#5!""$!2NLLLLLLL$!#;7$$"#5!""$!#X!""7$$"#5!""$!2lmmmmmmm#!#;7$$"#5!""$!#N!""7$$"#5!""$!2NLLLLLLL#!#;7$$"#5!""$!#D!""7$$"#5!""$!2nmmmmmmm"!#;7$$"#5!""$!2LLLLLLLL"!#;7$$"#?!""$!#S!""7$$"#?!""$!#I!""7$$"#?!""$!#?!""7$$"#?!""$!#5!""7$$"#?!""$!1LLLLLLL[!#:7$$"#?!""$!1nmmmmmmT!#:7$$"#?!""$!2NLLLLLL$Q!#;7$$"#?!""$!2lmmmmmm;$!#;7$$"#?!""$!2NLLLLLL$G!#;7$$"#?!""$!1nmmmmmmY!#:7$$"#?!""$!1LLLLLLLV!#:7$$"#?!""$!2lmmmmmm;#!#;7$$"#?!""$!2lmmmmmmm$!#;7$$"#?!""$!2NLLLLLLL$!#;7$$"#?!""$!#X!""7$$"#?!""$!2lmmmmmmm#!#;7$$"#?!""$!#N!""7$$"#?!""$!2NLLLLLLL#!#;7$$"#?!""$!#D!""7$$"#?!""$!2nmmmmmmm"!#;7$$"#?!""$!2LLLLLLLL"!#;7$$"#I!""$!#S!""7$$"#I!""$!#I!""7$$"#I!""$!#?!""7$$"#I!""$!1LLLLLLL[!#:7$$"#I!""$!1nmmmmmmT!#:7$$"#I!""$!2NLLLLLL$Q!#;7$$"#I!""$!2lmmmmmm;$!#;7$$"#I!""$!1nmmmmmmY!#:7$$"#I!""$!1LLLLLLLV!#:7$$"#I!""$!2lmmmmmmm$!#;7$$"#I!""$!2NLLLLLLL$!#;7$$"#I!""$!#X!""7$$"#I!""$!2lmmmmmmm#!#;7$$"#I!""$!#N!""7$$"#I!""$!2NLLLLLLL#!#;7$$"#S!""$!#S!""7$$"#S!""$!#I!""7$$"#S!""$!1LLLLLLL[!#:7$$"#S!""$!1nmmmmmmT!#:7$$"#S!""$!1nmmmmmmY!#:7$$"#S!""$!1LLLLLLLV!#:7$$"#S!""$!2lmmmmmmm$!#;7$$"#S!""$!2NLLLLLLL$!#;7$$"#S!""$!#X!""7$$"#]!""$!#S!""7$$"#]!""$!1nmmmmmmY!#:7$$"#]!""$!1LLLLLLLV!#:-%&COLORG6&%$RGBG$""!!""$""!!""$"#5!""-%'SYMBOLG6$%(DIAMONDG"#D-%'POINTSG6jp7$$""!!""$!#S!""7$$""!!""$!#I!""7$$""!!""$!#?!""7$$""!!""$!#5!""7$$""!!""$!1LLLLLLL[!#:7$$""!!""$!1nmmmmmmT!#:7$$""!!""$!2NLLLLLL$Q!#;7$$""!!""$!2lmmmmmm;$!#;7$$""!!""$!2NLLLLLL$G!#;7$$""!!""$!1nmmmmmmY!#:7$$""!!""$!1LLLLLLLV!#:7$$""!!""$!2lmmmmmm;#!#;7$$""!!""$!2lmmmmmmm$!#;7$$""!!""$!2LLLLLLL$=!#;7$$""!!""$!2NLLLLLLL$!#;7$$""!!""$!#X!""7$$""!!""$!2lmmmmmmm#!#;7$$""!!""$!#N!""7$$""!!""$!2NLLLLLLL#!#;7$$""!!""$!2nmmmmmm;"!#;7$$""!!""$!#D!""7$$""!!""$!2nmmmmmmm"!#;7$$""!!""$!2LLLLLLLL"!#;7$$""!!""$!#:!""7$$""!!""$!1mmmmmmmm!#;7$$""!!""$!1LLLLLLLL!#;7$$"#5!""$!#S!""7$$"#5!""$!#I!""7$$"#5!""$!#?!""7$$"#5!""$!1LLLLLLL[!#:7$$"#5!""$!1nmmmmmmT!#:7$$"#5!""$!2NLLLLLL$Q!#;7$$"#5!""$!2lmmmmmm;$!#;7$$"#5!""$!2NLLLLLL$G!#;7$$"#5!""$!1nmmmmmmY!#:7$$"#5!""$!1LLLLLLLV!#:7$$"#5!""$!2lmmmmmm;#!#;7$$"#5!""$!2lmmmmmmm$!#;7$$"#5!""$!2NLLLLLLL$!#;7$$"#5!""$!#X!""7$$"#5!""$!2lmmmmmmm#!#;7$$"#5!""$!#N!""7$$"#5!""$!2NLLLLLLL#!#;7$$"#5!""$!#D!""7$$"#5!""$!2nmmmmmmm"!#;7$$"#5!""$!2LLLLLLLL"!#;7$$"#?!""$!#S!""7$$"#?!""$!#I!""7$$"#?!""$!#?!""7$$"#?!""$!1LLLLLLL[!#:7$$"#?!""$!1nmmmmmmT!#:7$$"#?!""$!2NLLLLLL$Q!#;7$$"#?!""$!2lmmmmmm;$!#;7$$"#?!""$!2NLLLLLL$G!#;7$$"#?!""$!1nmmmmmmY!#:7$$"#?!""$!1LLLLLLLV!#:7$$"#?!""$!2lmmmmmm;#!#;7$$"#?!""$!2lmmmmmmm$!#;7$$"#?!""$!2NLLLLLLL$!#;7$$"#?!""$!#X!""7$$"#?!""$!2lmmmmmmm#!#;7$$"#?!""$!#N!""7$$"#?!""$!2NLLLLLLL#!#;7$$"#?!""$!#D!""7$$"#?!""$!2nmmmmmmm"!#;7$$"#?!""$!2LLLLLLLL"!#;7$$"#I!""$!#S!""7$$"#I!""$!#I!""7$$"#I!""$!1LLLLLLL[!#:7$$"#I!""$!1nmmmmmmT!#:7$$"#I!""$!2NLLLLLL$Q!#;7$$"#I!""$!2lmmmmmm;$!#;7$$"#I!""$!1nmmmmmmY!#:7$$"#I!""$!1LLLLLLLV!#:7$$"#I!""$!2lmmmmmmm$!#;7$$"#I!""$!2NLLLLLLL$!#;7$$"#I!""$!#X!""7$$"#I!""$!2lmmmmmmm#!#;7$$"#I!""$!#N!""7$$"#I!""$!2NLLLLLLL#!#;7$$"#S!""$!#S!""7$$"#S!""$!#I!""7$$"#S!""$!1LLLLLLL[!#:7$$"#S!""$!1nmmmmmmT!#:7$$"#S!""$!1nmmmmmmY!#:7$$"#S!""$!1LLLLLLLV!#:7$$"#S!""$!2lmmmmmmm$!#;7$$"#S!""$!2NLLLLLLL$!#;7$$"#S!""$!#X!""7$$"#]!""$!#S!""7$$"#]!""$!1nmmmmmmY!#:7$$"#]!""$!1LLLLLLLV!#:-%&COLORG6&%$RGBG$"#5!""$""!!""$""!!""-%'SYMBOLG6$%(DIAMONDG"#D-%'POINTSG6hp7$$""!!""$!#S!""7$$""!!""$!#I!""7$$""!!""$!#?!""7$$""!!""$!#5!""7$$""!!""$!1LLLLLLL[!#:7$$""!!""$!1nmmmmmmT!#:7$$""!!""$!2NLLLLLL$Q!#;7$$""!!""$!2lmmmmmm;$!#;7$$""!!""$!2NLLLLLL$G!#;7$$""!!""$!1nmmmmmmY!#:7$$""!!""$!1LLLLLLLV!#:7$$""!!""$!2lmmmmmm;#!#;7$$""!!""$!2lmmmmmmm$!#;7$$""!!""$!2LLLLLLL$=!#;7$$""!!""$!2NLLLLLLL$!#;7$$""!!""$!#X!""7$$""!!""$!2lmmmmmmm#!#;7$$""!!""$!#N!""7$$""!!""$!2NLLLLLLL#!#;7$$""!!""$!2nmmmmmm;"!#;7$$""!!""$!#D!""7$$""!!""$!2nmmmmmmm"!#;7$$""!!""$!2LLLLLLLL"!#;7$$""!!""$!#:!""7$$""!!""$!1mmmmmmmm!#;7$$"#5!""$!#S!""7$$"#5!""$!#I!""7$$"#5!""$!#?!""7$$"#5!""$!1LLLLLLL[!#:7$$"#5!""$!1nmmmmmmT!#:7$$"#5!""$!2NLLLLLL$Q!#;7$$"#5!""$!2lmmmmmm;$!#;7$$"#5!""$!2NLLLLLL$G!#;7$$"#5!""$!1nmmmmmmY!#:7$$"#5!""$!1LLLLLLLV!#:7$$"#5!""$!2lmmmmmm;#!#;7$$"#5!""$!2lmmmmmmm$!#;7$$"#5!""$!2NLLLLLLL$!#;7$$"#5!""$!#X!""7$$"#5!""$!2lmmmmmmm#!#;7$$"#5!""$!#N!""7$$"#5!""$!2NLLLLLLL#!#;7$$"#5!""$!#D!""7$$"#5!""$!2nmmmmmmm"!#;7$$"#5!""$!2LLLLLLLL"!#;7$$"#?!""$!#S!""7$$"#?!""$!#I!""7$$"#?!""$!#?!""7$$"#?!""$!1LLLLLLL[!#:7$$"#?!""$!1nmmmmmmT!#:7$$"#?!""$!2NLLLLLL$Q!#;7$$"#?!""$!2lmmmmmm;$!#;7$$"#?!""$!2NLLLLLL$G!#;7$$"#?!""$!1nmmmmmmY!#:7$$"#?!""$!1LLLLLLLV!#:7$$"#?!""$!2lmmmmmm;#!#;7$$"#?!""$!2lmmmmmmm$!#;7$$"#?!""$!2NLLLLLLL$!#;7$$"#?!""$!#X!""7$$"#?!""$!2lmmmmmmm#!#;7$$"#?!""$!#N!""7$$"#?!""$!2NLLLLLLL#!#;7$$"#?!""$!#D!""7$$"#?!""$!2nmmmmmmm"!#;7$$"#I!""$!#S!""7$$"#I!""$!#I!""7$$"#I!""$!1LLLLLLL[!#:7$$"#I!""$!1nmmmmmmT!#:7$$"#I!""$!2NLLLLLL$Q!#;7$$"#I!""$!2lmmmmmm;$!#;7$$"#I!""$!1nmmmmmmY!#:7$$"#I!""$!1LLLLLLLV!#:7$$"#I!""$!2lmmmmmmm$!#;7$$"#I!""$!2NLLLLLLL$!#;7$$"#I!""$!#X!""7$$"#I!""$!2lmmmmmmm#!#;7$$"#I!""$!#N!""7$$"#I!""$!2NLLLLLLL#!#;7$$"#S!""$!#S!""7$$"#S!""$!#I!""7$$"#S!""$!1LLLLLLL[!#:7$$"#S!""$!1nmmmmmmT!#:7$$"#S!""$!1nmmmmmmY!#:7$$"#S!""$!1LLLLLLLV!#:7$$"#S!""$!2lmmmmmmm$!#;7$$"#S!""$!2NLLLLLLL$!#;7$$"#S!""$!#X!""7$$"#]!""$!#S!""7$$"#]!""$!1nmmmmmmY!#:7$$"#]!""$!1LLLLLLLV!#:-%&COLORG6&%$RGBG$""!!""$"#5!""$""!!""-%'SYMBOLG6$%(DIAMONDG"#D-%%VIEWG6$;$""!!""$"#]!"";$!1LLLLLLLV!#:$""!!""-&%&_AXISG6#"""6(-%+_GRIDLINESG6(-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#""!-%%FONTG6%%*HELVETICAG%&ROMANG"#:-%-TRANSPARENCYG6#$""!!""-%)_VISIBLEG6#""!-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#""!-%%FONTG6%%*HELVETICAG%&ROMANG"#:-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6(-%+_GRIDLINESG6(-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#""!-%%FONTG6%%*HELVETICAG%&ROMANG"#:-%-TRANSPARENCYG6#$""!!""-%)_VISIBLEG6#""!-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#""!-%%FONTG6%%*HELVETICAG%&ROMANG"#:-%-TRANSPARENCYG6#$""!!""-%*AXESTICKSG6%%(DEFAULTG%(DEFAULTG-%%FONTG6%%*HELVETICAG%&ROMANG"#:-%+AXESLABELSG6%Q"i6"Q"j6"-%%FONTG6%%*HELVETICAG%(DEFAULTG"#=-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$"$g&!""-%)BOUNDS_YG6#$"$g"!""-%-BOUNDS_WIDTHG6#$"%IL!""-%.BOUNDS_HEIGHTG6#$"%?P!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+S!""-%.BOUNDS_HEIGHTG6#$"%+S!""-%)CHILDRENG6"G6"
at this point we find our choice for c,
should we also set q[2]=-2/3??? (Check later if this is really necessary)
csubs;
factor(getel(posFfab,0,-2/3));factor(subs(csubs,%));
factor(getel(posFfab,1,-4/3));factor(subs(csubs,%));
factor(getel(posFfab,2,-5/3));factor(subs(csubs,%));
L0kiY0c2IiomSSNhMUdGJCIiIikiIiMjRikiIiRGJw==
LCQqKiIiJSIiIkkma2FwcGFHNiJGJSwmKiZJI2ExR0YnRiUpIiIjI0YsIiIkRiVGJUkiY0dGJyEiIkYlKUkibkdGJ0YtRjBGJQ==
IiIh
LCQqLiMiIzsiIiQiIiIpIiIjI0YnRiZGJ0knbGFtYmRhRzYiRidJIm1HRixGJywmKiZGJkYnJkkicUdGLDYjRilGJ0YnRilGJ0YnKUkibkdGLCMiIiVGJiEiIkYn
LCQqLiMiIzsiIiQiIiIpIiIjI0YnRiZGJ0knbGFtYmRhRzYiRidJIm1HRixGJywmKiZGJkYnJkkicUdGLDYjRilGJ0YnRilGJ0YnKUkibkdGLCMiIiVGJiEiIkYn
LCQqLiIiJSIiIkkma2FwcGFHNiJGJSlJIm1HRiciIiNGJSZJInFHRic2I0YqRiUsJiomSSNhMUdGJ0YlKUYqI0YqIiIkRiVGJUkiY0dGJyEiIkYlKUkibkdGJyMiIiZGM0Y1RiU=
IiIh
set a=2, b=0, c=a1*2^(2/3),
q[2]=-2/3
posFfabc := expand(simplify(subs(csubs,q[2]=-2/3,posFfab))) assuming n::posint,m::posint:
newtabc := mynewt(posFfabc,m,n):
P1abc := pointplot(newtabc,myoptionsLo,color=yellow):
display(P1,P1a,P1ab,P1abc,myview);
6.-%'POINTSG6^q7$$""!!""$!#S!""7$$""!!""$!#I!""7$$""!!""$!#?!""7$$""!!""$!#5!""7$$""!!""$""!!""7$$""!!""$!1LLLLLLL[!#:7$$""!!""$!1nmmmmmmT!#:7$$""!!""$!2NLLLLLL$Q!#;7$$""!!""$!2lmmmmmm;$!#;7$$""!!""$!2NLLLLLL$G!#;7$$""!!""$!1nmmmmmmY!#:7$$""!!""$!1LLLLLLLV!#:7$$""!!""$!2lmmmmmm;#!#;7$$""!!""$!2lmmmmmmm$!#;7$$""!!""$!2LLLLLLL$=!#;7$$""!!""$!2NLLLLLLL$!#;7$$""!!""$!#X!""7$$""!!""$!2lmmmmmmm#!#;7$$""!!""$!#N!""7$$""!!""$!2NLLLLLLL#!#;7$$""!!""$!2nmmmmmm;"!#;7$$""!!""$!#D!""7$$""!!""$!2nmmmmmmm"!#;7$$""!!""$!2LLLLLLLL"!#;7$$""!!""$!#:!""7$$""!!""$!1mmmmmmmm!#;7$$""!!""$!1LLLLLLLL!#;7$$"#5!""$!#S!""7$$"#5!""$!#I!""7$$"#5!""$!#?!""7$$"#5!""$!#5!""7$$"#5!""$!1LLLLLLL[!#:7$$"#5!""$!1nmmmmmmT!#:7$$"#5!""$!2NLLLLLL$Q!#;7$$"#5!""$!2lmmmmmm;$!#;7$$"#5!""$!2NLLLLLL$G!#;7$$"#5!""$!1nmmmmmmY!#:7$$"#5!""$!1LLLLLLLV!#:7$$"#5!""$!2lmmmmmm;#!#;7$$"#5!""$!2lmmmmmmm$!#;7$$"#5!""$!2NLLLLLLL$!#;7$$"#5!""$!#X!""7$$"#5!""$!2lmmmmmmm#!#;7$$"#5!""$!#N!""7$$"#5!""$!2NLLLLLLL#!#;7$$"#5!""$!#D!""7$$"#5!""$!2nmmmmmmm"!#;7$$"#5!""$!2LLLLLLLL"!#;7$$"#?!""$!#S!""7$$"#?!""$!#I!""7$$"#?!""$!#?!""7$$"#?!""$!#5!""7$$"#?!""$!1LLLLLLL[!#:7$$"#?!""$!1nmmmmmmT!#:7$$"#?!""$!2NLLLLLL$Q!#;7$$"#?!""$!2lmmmmmm;$!#;7$$"#?!""$!2NLLLLLL$G!#;7$$"#?!""$!1nmmmmmmY!#:7$$"#?!""$!1LLLLLLLV!#:7$$"#?!""$!2lmmmmmm;#!#;7$$"#?!""$!2lmmmmmmm$!#;7$$"#?!""$!2NLLLLLLL$!#;7$$"#?!""$!#X!""7$$"#?!""$!2lmmmmmmm#!#;7$$"#?!""$!#N!""7$$"#?!""$!2NLLLLLLL#!#;7$$"#?!""$!#D!""7$$"#?!""$!2nmmmmmmm"!#;7$$"#?!""$!2LLLLLLLL"!#;7$$"#I!""$!#S!""7$$"#I!""$!#I!""7$$"#I!""$!#?!""7$$"#I!""$!1LLLLLLL[!#:7$$"#I!""$!1nmmmmmmT!#:7$$"#I!""$!2NLLLLLL$Q!#;7$$"#I!""$!2lmmmmmm;$!#;7$$"#I!""$!1nmmmmmmY!#:7$$"#I!""$!1LLLLLLLV!#:7$$"#I!""$!2lmmmmmmm$!#;7$$"#I!""$!2NLLLLLLL$!#;7$$"#I!""$!#X!""7$$"#I!""$!2lmmmmmmm#!#;7$$"#I!""$!#N!""7$$"#I!""$!2NLLLLLLL#!#;7$$"#S!""$!#S!""7$$"#S!""$!#I!""7$$"#S!""$!1LLLLLLL[!#:7$$"#S!""$!1nmmmmmmT!#:7$$"#S!""$!1nmmmmmmY!#:7$$"#S!""$!1LLLLLLLV!#:7$$"#S!""$!2lmmmmmmm$!#;7$$"#S!""$!2NLLLLLLL$!#;7$$"#S!""$!#X!""7$$"#]!""$!#S!""7$$"#]!""$!1nmmmmmmY!#:7$$"#]!""$!1LLLLLLLV!#:-%&COLORG6&%$RGBG$""!!""$""!!""$"#5!""-%'SYMBOLG6$%(DIAMONDG"#D-%'POINTSG6jp7$$""!!""$!#S!""7$$""!!""$!#I!""7$$""!!""$!#?!""7$$""!!""$!#5!""7$$""!!""$!1LLLLLLL[!#:7$$""!!""$!1nmmmmmmT!#:7$$""!!""$!2NLLLLLL$Q!#;7$$""!!""$!2lmmmmmm;$!#;7$$""!!""$!2NLLLLLL$G!#;7$$""!!""$!1nmmmmmmY!#:7$$""!!""$!1LLLLLLLV!#:7$$""!!""$!2lmmmmmm;#!#;7$$""!!""$!2lmmmmmmm$!#;7$$""!!""$!2LLLLLLL$=!#;7$$""!!""$!2NLLLLLLL$!#;7$$""!!""$!#X!""7$$""!!""$!2lmmmmmmm#!#;7$$""!!""$!#N!""7$$""!!""$!2NLLLLLLL#!#;7$$""!!""$!2nmmmmmm;"!#;7$$""!!""$!#D!""7$$""!!""$!2nmmmmmmm"!#;7$$""!!""$!2LLLLLLLL"!#;7$$""!!""$!#:!""7$$""!!""$!1mmmmmmmm!#;7$$""!!""$!1LLLLLLLL!#;7$$"#5!""$!#S!""7$$"#5!""$!#I!""7$$"#5!""$!#?!""7$$"#5!""$!1LLLLLLL[!#:7$$"#5!""$!1nmmmmmmT!#:7$$"#5!""$!2NLLLLLL$Q!#;7$$"#5!""$!2lmmmmmm;$!#;7$$"#5!""$!2NLLLLLL$G!#;7$$"#5!""$!1nmmmmmmY!#:7$$"#5!""$!1LLLLLLLV!#:7$$"#5!""$!2lmmmmmm;#!#;7$$"#5!""$!2lmmmmmmm$!#;7$$"#5!""$!2NLLLLLLL$!#;7$$"#5!""$!#X!""7$$"#5!""$!2lmmmmmmm#!#;7$$"#5!""$!#N!""7$$"#5!""$!2NLLLLLLL#!#;7$$"#5!""$!#D!""7$$"#5!""$!2nmmmmmmm"!#;7$$"#5!""$!2LLLLLLLL"!#;7$$"#?!""$!#S!""7$$"#?!""$!#I!""7$$"#?!""$!#?!""7$$"#?!""$!1LLLLLLL[!#:7$$"#?!""$!1nmmmmmmT!#:7$$"#?!""$!2NLLLLLL$Q!#;7$$"#?!""$!2lmmmmmm;$!#;7$$"#?!""$!2NLLLLLL$G!#;7$$"#?!""$!1nmmmmmmY!#:7$$"#?!""$!1LLLLLLLV!#:7$$"#?!""$!2lmmmmmm;#!#;7$$"#?!""$!2lmmmmmmm$!#;7$$"#?!""$!2NLLLLLLL$!#;7$$"#?!""$!#X!""7$$"#?!""$!2lmmmmmmm#!#;7$$"#?!""$!#N!""7$$"#?!""$!2NLLLLLLL#!#;7$$"#?!""$!#D!""7$$"#?!""$!2nmmmmmmm"!#;7$$"#?!""$!2LLLLLLLL"!#;7$$"#I!""$!#S!""7$$"#I!""$!#I!""7$$"#I!""$!1LLLLLLL[!#:7$$"#I!""$!1nmmmmmmT!#:7$$"#I!""$!2NLLLLLL$Q!#;7$$"#I!""$!2lmmmmmm;$!#;7$$"#I!""$!1nmmmmmmY!#:7$$"#I!""$!1LLLLLLLV!#:7$$"#I!""$!2lmmmmmmm$!#;7$$"#I!""$!2NLLLLLLL$!#;7$$"#I!""$!#X!""7$$"#I!""$!2lmmmmmmm#!#;7$$"#I!""$!#N!""7$$"#I!""$!2NLLLLLLL#!#;7$$"#S!""$!#S!""7$$"#S!""$!#I!""7$$"#S!""$!1LLLLLLL[!#:7$$"#S!""$!1nmmmmmmT!#:7$$"#S!""$!1nmmmmmmY!#:7$$"#S!""$!1LLLLLLLV!#:7$$"#S!""$!2lmmmmmmm$!#;7$$"#S!""$!2NLLLLLLL$!#;7$$"#S!""$!#X!""7$$"#]!""$!#S!""7$$"#]!""$!1nmmmmmmY!#:7$$"#]!""$!1LLLLLLLV!#:-%&COLORG6&%$RGBG$"#5!""$""!!""$""!!""-%'SYMBOLG6$%(DIAMONDG"#D-%'POINTSG6hp7$$""!!""$!#S!""7$$""!!""$!#I!""7$$""!!""$!#?!""7$$""!!""$!#5!""7$$""!!""$!1LLLLLLL[!#:7$$""!!""$!1nmmmmmmT!#:7$$""!!""$!2NLLLLLL$Q!#;7$$""!!""$!2lmmmmmm;$!#;7$$""!!""$!2NLLLLLL$G!#;7$$""!!""$!1nmmmmmmY!#:7$$""!!""$!1LLLLLLLV!#:7$$""!!""$!2lmmmmmm;#!#;7$$""!!""$!2lmmmmmmm$!#;7$$""!!""$!2LLLLLLL$=!#;7$$""!!""$!2NLLLLLLL$!#;7$$""!!""$!#X!""7$$""!!""$!2lmmmmmmm#!#;7$$""!!""$!#N!""7$$""!!""$!2NLLLLLLL#!#;7$$""!!""$!2nmmmmmm;"!#;7$$""!!""$!#D!""7$$""!!""$!2nmmmmmmm"!#;7$$""!!""$!2LLLLLLLL"!#;7$$""!!""$!#:!""7$$""!!""$!1mmmmmmmm!#;7$$"#5!""$!#S!""7$$"#5!""$!#I!""7$$"#5!""$!#?!""7$$"#5!""$!1LLLLLLL[!#:7$$"#5!""$!1nmmmmmmT!#:7$$"#5!""$!2NLLLLLL$Q!#;7$$"#5!""$!2lmmmmmm;$!#;7$$"#5!""$!2NLLLLLL$G!#;7$$"#5!""$!1nmmmmmmY!#:7$$"#5!""$!1LLLLLLLV!#:7$$"#5!""$!2lmmmmmm;#!#;7$$"#5!""$!2lmmmmmmm$!#;7$$"#5!""$!2NLLLLLLL$!#;7$$"#5!""$!#X!""7$$"#5!""$!2lmmmmmmm#!#;7$$"#5!""$!#N!""7$$"#5!""$!2NLLLLLLL#!#;7$$"#5!""$!#D!""7$$"#5!""$!2nmmmmmmm"!#;7$$"#5!""$!2LLLLLLLL"!#;7$$"#?!""$!#S!""7$$"#?!""$!#I!""7$$"#?!""$!#?!""7$$"#?!""$!1LLLLLLL[!#:7$$"#?!""$!1nmmmmmmT!#:7$$"#?!""$!2NLLLLLL$Q!#;7$$"#?!""$!2lmmmmmm;$!#;7$$"#?!""$!2NLLLLLL$G!#;7$$"#?!""$!1nmmmmmmY!#:7$$"#?!""$!1LLLLLLLV!#:7$$"#?!""$!2lmmmmmm;#!#;7$$"#?!""$!2lmmmmmmm$!#;7$$"#?!""$!2NLLLLLLL$!#;7$$"#?!""$!#X!""7$$"#?!""$!2lmmmmmmm#!#;7$$"#?!""$!#N!""7$$"#?!""$!2NLLLLLLL#!#;7$$"#?!""$!#D!""7$$"#?!""$!2nmmmmmmm"!#;7$$"#I!""$!#S!""7$$"#I!""$!#I!""7$$"#I!""$!1LLLLLLL[!#:7$$"#I!""$!1nmmmmmmT!#:7$$"#I!""$!2NLLLLLL$Q!#;7$$"#I!""$!2lmmmmmm;$!#;7$$"#I!""$!1nmmmmmmY!#:7$$"#I!""$!1LLLLLLLV!#:7$$"#I!""$!2lmmmmmmm$!#;7$$"#I!""$!2NLLLLLLL$!#;7$$"#I!""$!#X!""7$$"#I!""$!2lmmmmmmm#!#;7$$"#I!""$!#N!""7$$"#I!""$!2NLLLLLLL#!#;7$$"#S!""$!#S!""7$$"#S!""$!#I!""7$$"#S!""$!1LLLLLLL[!#:7$$"#S!""$!1nmmmmmmT!#:7$$"#S!""$!1nmmmmmmY!#:7$$"#S!""$!1LLLLLLLV!#:7$$"#S!""$!2lmmmmmmm$!#;7$$"#S!""$!2NLLLLLLL$!#;7$$"#S!""$!#X!""7$$"#]!""$!#S!""7$$"#]!""$!1nmmmmmmY!#:7$$"#]!""$!1LLLLLLLV!#:-%&COLORG6&%$RGBG$""!!""$"#5!""$""!!""-%'SYMBOLG6$%(DIAMONDG"#D-%'POINTSG6ep7$$""!!""$!#S!""7$$""!!""$!#I!""7$$""!!""$!#?!""7$$""!!""$!#5!""7$$""!!""$!1LLLLLLL[!#:7$$""!!""$!1nmmmmmmT!#:7$$""!!""$!2NLLLLLL$Q!#;7$$""!!""$!2lmmmmmm;$!#;7$$""!!""$!2NLLLLLL$G!#;7$$""!!""$!1nmmmmmmY!#:7$$""!!""$!1LLLLLLLV!#:7$$""!!""$!2lmmmmmm;#!#;7$$""!!""$!2lmmmmmmm$!#;7$$""!!""$!2LLLLLLL$=!#;7$$""!!""$!2NLLLLLLL$!#;7$$""!!""$!#X!""7$$""!!""$!2lmmmmmmm#!#;7$$""!!""$!#N!""7$$""!!""$!2NLLLLLLL#!#;7$$""!!""$!2nmmmmmm;"!#;7$$""!!""$!#D!""7$$""!!""$!2nmmmmmmm"!#;7$$""!!""$!2LLLLLLLL"!#;7$$""!!""$!#:!""7$$"#5!""$!#S!""7$$"#5!""$!#I!""7$$"#5!""$!#?!""7$$"#5!""$!1LLLLLLL[!#:7$$"#5!""$!1nmmmmmmT!#:7$$"#5!""$!2NLLLLLL$Q!#;7$$"#5!""$!2lmmmmmm;$!#;7$$"#5!""$!2NLLLLLL$G!#;7$$"#5!""$!1nmmmmmmY!#:7$$"#5!""$!1LLLLLLLV!#:7$$"#5!""$!2lmmmmmm;#!#;7$$"#5!""$!2lmmmmmmm$!#;7$$"#5!""$!2NLLLLLLL$!#;7$$"#5!""$!#X!""7$$"#5!""$!2lmmmmmmm#!#;7$$"#5!""$!#N!""7$$"#5!""$!2NLLLLLLL#!#;7$$"#5!""$!#D!""7$$"#5!""$!2nmmmmmmm"!#;7$$"#?!""$!#S!""7$$"#?!""$!#I!""7$$"#?!""$!#?!""7$$"#?!""$!1LLLLLLL[!#:7$$"#?!""$!1nmmmmmmT!#:7$$"#?!""$!2NLLLLLL$Q!#;7$$"#?!""$!2lmmmmmm;$!#;7$$"#?!""$!2NLLLLLL$G!#;7$$"#?!""$!1nmmmmmmY!#:7$$"#?!""$!1LLLLLLLV!#:7$$"#?!""$!2lmmmmmm;#!#;7$$"#?!""$!2lmmmmmmm$!#;7$$"#?!""$!2NLLLLLLL$!#;7$$"#?!""$!#X!""7$$"#?!""$!2lmmmmmmm#!#;7$$"#?!""$!#N!""7$$"#?!""$!2NLLLLLLL#!#;7$$"#?!""$!#D!""7$$"#I!""$!#S!""7$$"#I!""$!#I!""7$$"#I!""$!1LLLLLLL[!#:7$$"#I!""$!1nmmmmmmT!#:7$$"#I!""$!2NLLLLLL$Q!#;7$$"#I!""$!2lmmmmmm;$!#;7$$"#I!""$!1nmmmmmmY!#:7$$"#I!""$!1LLLLLLLV!#:7$$"#I!""$!2lmmmmmmm$!#;7$$"#I!""$!2NLLLLLLL$!#;7$$"#I!""$!#X!""7$$"#I!""$!2lmmmmmmm#!#;7$$"#I!""$!#N!""7$$"#I!""$!2NLLLLLLL#!#;7$$"#S!""$!#S!""7$$"#S!""$!#I!""7$$"#S!""$!1LLLLLLL[!#:7$$"#S!""$!1nmmmmmmT!#:7$$"#S!""$!1nmmmmmmY!#:7$$"#S!""$!1LLLLLLLV!#:7$$"#S!""$!2lmmmmmmm$!#;7$$"#S!""$!2NLLLLLLL$!#;7$$"#S!""$!#X!""7$$"#]!""$!#S!""7$$"#]!""$!1nmmmmmmY!#:7$$"#]!""$!1LLLLLLLV!#:-%&COLORG6&%$RGBG$"#5!""$"#5!""$""!!""-%'SYMBOLG6$%(DIAMONDG"#D-%%VIEWG6$;$""!!""$"#]!"";$!1LLLLLLLV!#:$""!!""-&%&_AXISG6#"""6(-%+_GRIDLINESG6(-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#""!-%%FONTG6%%*HELVETICAG%&ROMANG"#:-%-TRANSPARENCYG6#$""!!""-%)_VISIBLEG6#""!-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#""!-%%FONTG6%%*HELVETICAG%&ROMANG"#:-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6(-%+_GRIDLINESG6(-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#""!-%%FONTG6%%*HELVETICAG%&ROMANG"#:-%-TRANSPARENCYG6#$""!!""-%)_VISIBLEG6#""!-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#""!-%%FONTG6%%*HELVETICAG%&ROMANG"#:-%-TRANSPARENCYG6#$""!!""-%*AXESTICKSG6%%(DEFAULTG%(DEFAULTG-%%FONTG6%%*HELVETICAG%&ROMANG"#:-%+AXESLABELSG6%Q"i6"Q"j6"-%%FONTG6%%*HELVETICAG%(DEFAULTG"#=-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$"$g&!""-%)BOUNDS_YG6#$"$g"!""-%-BOUNDS_WIDTHG6#$"%IL!""-%.BOUNDS_HEIGHTG6#$"%?P!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+S!""-%.BOUNDS_HEIGHTG6#$"%+S!""-%)CHILDRENG6"Ig==
Here we get pterm
factor(getel(posFfabc,0,-1));
factor(getel(posFfabc,1,-4/3));
solve({%%,%},{q[2],pterm});
LCQqKiMiIiIiIiRGJUkma2FwcGFHNiJGJSwmIiNIISIiKiYiIzdGJUkmcHRlcm1HRihGJUYlRiVJIm5HRihGK0Yr
IiIh
PCQvSSZwdGVybUc2IiMiI0giIzcvJkkicUdGJTYjIiIjRio=
set a=2, b=0,c,pterm=29/12,q[2]=-2/3
posFfabcd := expand(simplify(subs(pterm=29/12,q[2]=-2/3,posFfabc))) assuming n::posint,m::posint:
newtabcd := mynewt(posFfabcd,m,n):
only the brown points remain
Now all points are strictly below n^{-1}
P1abcd := pointplot(newtabcd,myoptionsLo,color=brown):
display(P1,P1a,P1ab,P1abc,P1abcd,myview);
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Here are the dominating corners and we see that we have to choose d=-1 to have a positive term;
note that we will see that the second term should be negative, as lambda=Ai' is negative for large m
getel(posFfabcd,0,-7/6);
getel(posFfabcd,3,-14/6);
(14/6-7/6)/3; #slope
LCQqKiIiJSIiIilJIm5HNiIjIiIoIiInISIiSSZrYXBwYUdGKEYlSSJkR0YoRiVGLA==
LCQqLCMiI2siIioiIiIpSSJuRzYiIyIiKCIiJCEiIikiIiMjRidGLUYnSSdsYW1iZGFHRipGJylJIm1HRipGLUYnRi4=
IyIiKCIjPQ==
and continuing
getel(posFfabcd,4,-3);
(3-14/6)/1; #slope
LCQqKiMiJE8iIiIqIiIiKUkibkc2IiIiJCEiIkkma2FwcGFHRipGJylJIm1HRioiIiVGJ0Ys
IyIiIyIiJA==
set a=2, b=0,c,pterm=13/6,q[2]=-2/3 and d=-1
posFfabcde := expand(simplify(subs(d=-1,posFfabcd))) assuming n::posint,m::posint:
newtabcde := mynewt(posFfabcde,m,n):
This is the final result, where only the solid diamonds are non-zero
P1abcde := pointplot(newtabcde,myoptionsLo,symbol=soliddiamond,color=gray):
display(P1,P1a,P1ab,P1abc,P1abcd,P1abcde,myview);
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Plot the boundary and the slopes of the Newton polygon;
Note that we have already proved that there are now points above the blue dotted line
LegendSize := size=[600,450]:
P1dom1 := plot(-7/6-(7/18)*m,m=0..3,color=black,legend=[typeset("Slope: ", -7/18)],legendstyle=[location=right]):
P1dom2 := plot(-1/3-(2/3)*m,m=3..4,color=red,legend=[typeset("Slope: ", -2/3)],legendstyle=[location=right]):
P1dom3a := plot(1-m,m=0..5,color=blue,linestyle=dot,legend=[typeset("Slope: ", -1)],legendstyle=[location=right]):
P1all := display(P1,P1a,P1ab,P1abc,P1abcd,P1abcde,P1dom1,P1dom2,P1dom3a,myview,LegendSize):
for i from 1 to 8 do
P1dom3[i] := plot([[1+i/2,0],[1+i/2,-i/2]],color=blue,linestyle=dot):
end:
display(P1all,seq(P1dom3[i],i=1..8));
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This is the choice for SF
SF(n);
subs(a=2,b=0,csubs,pterm=29/12,d=-1,%);
LCxJImFHNiIiIiIqJkkiYkdGJEYlKUkibkdGJCNGJSIiJCEiIkYlKiZJImNHRiRGJSlGKSMiIiNGK0YsRiUqJkkmcHRlcm1HRiRGJUYpRixGJSomSSJkR0YkRiUpRikjIiIoIiInRixGJQ==
LCoiIiMiIiIqKEkjYTFHNiJGJClGIyNGIyIiJEYkKUkibkdGJ0YpISIiRiQqJiMiI0giIzdGJEYsRi1GJCokKUYsIyIiKCIiJ0YtRi0=
recall
kaplam;
NiQvLUknQWlyeUFpRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJi1JLEFpcnlBaVplcm9zR0YmNiMiIiJGLyooKSIiIyNGLyIiJEYvSSJtR0YpRi8pSSJuR0YpRjMhIiJGL0kma2FwcGFHRikvLUYlNiRGL0YrSSdsYW1iZGFHRik=
Look at the corners
Here we see the necessary choice: p[4]>2/9 (note that for m>n^(1/3) lambda is negative)
getel(posFfabcde,0,-7/6);
simplify(getel(posFfabcde,3,-14/6));
simplify(getel(posFfabcd,4,-3));
simplify(getel(posFfabcd,5,-4));
#simplify(getel(posFfabcd,6,-5));
LCQqKCIiJSIiIilJIm5HNiIjIiIoIiInISIiSSZrYXBwYUdGKEYlRiU=
LCQqLCMiI2siIioiIiIpSSJuRzYiIyIiKCIiJCEiIikiIiMjRidGLUYnSSdsYW1iZGFHRipGJylJIm1HRipGLUYnRi4=
LCQqKiMiJE8iIiIqIiIiKUkibkc2IiIiJCEiIkkma2FwcGFHRipGJylJIm1HRioiIiVGJ0Ys
LCQqKiMiJFsjIiROIiIiIilJIm5HNiIiIiUhIiJJJmthcHBhR0YqRicpSSJtR0YqIiImRidGLA==
Now split the black dots into the contributions from Ai and Ai'
indets(posFfabcd);
PD5JI2ExRzYiSSJkR0YkSSZrYXBwYUdGJEknbGFtYmRhR0YkSSJtR0YkSSJuR0YkJkkicUdGJDYjIiIhJkYrNiMiIiIqJClGKSMiI0giIichIiIqJClGKSMiI0RGNUY2KiQpRikjIiNCRjVGNiokKUYpIyIjPkY1RjYqJClGKSMiIzxGNUY2KiQpRikjIiM5IiIkRjYqJClGKSMiIzhGS0Y2KiQpRikjRk9GNUY2KiQpRikjIiM2RktGNiokKUYpI0ZWRjVGNiokKUYpIyIjNUZLRjYqJClGKSMiIioiIiNGNiokKUYpIyIiKUZLRjYqJClGKSMiIihGXG9GNiokKUYpI0Zkb0ZLRjYqJClGKSNGZG9GNUY2KiQpRikjIiImRlxvRjYqJClGKSNGXnBGS0Y2KiQpRikjIiIlRktGNiokKUYpI0ZLRlxvRjY=
Sanity check that there are no other contributions
subs(kappa=0,lambda=0,posFfabcd);
IiIh
Extract the coefficients of kappa=Ai and lambda=Ai'
and treat then seperately
posFk := coeff(posFfabcd,kappa):indets(%);
posFl := coeff(posFfabcd,lambda):indets(%);
PDtJI2ExRzYiSSJkR0YkSSJtR0YkSSJuR0YkJkkicUdGJDYjIiIhJkYpNiMiIiIqJClGJyMiI0giIichIiIqJClGJyMiI0RGM0Y0KiQpRicjIiNCRjNGNCokKUYnIyIjPkYzRjQqJClGJyMiIzxGM0Y0KiQpRicjIiM5IiIkRjQqJClGJyMiIzhGSUY0KiQpRicjRk1GM0Y0KiQpRicjIiM2RklGNCokKUYnI0ZURjNGNCokKUYnIyIjNUZJRjQqJClGJyMiIioiIiNGNCokKUYnIyIiKUZJRjQqJClGJyMiIihGam5GNCokKUYnI0Zib0ZJRjQqJClGJyNGYm9GM0Y0KiQpRicjIiImRmpuRjQqJClGJyNGXHBGSUY0KiQpRicjIiIlRklGNA==
PDpJI2ExRzYiSSJkR0YkSSJtR0YkSSJuR0YkJkkicUdGJDYjIiIhJkYpNiMiIiIqJClGJyMiI0giIichIiIqJClGJyMiI0RGM0Y0KiQpRicjIiNCRjNGNCokKUYnIyIjPkYzRjQqJClGJyMiIzxGM0Y0KiQpRicjIiM5IiIkRjQqJClGJyMiIzhGSUY0KiQpRicjRk1GM0Y0KiQpRicjIiM2RklGNCokKUYnIyIjNUZJRjQqJClGJyMiIioiIiNGNCokKUYnIyIiKUZJRjQqJClGJyMiIihGZ25GNCokKUYnI0Zfb0ZJRjQqJClGJyMiIiZGZ25GNCokKUYnI0Zmb0ZJRjQqJClGJyMiIiVGSUY0KiQpRicjRklGZ25GNA==
We color the non-zero nodes of the lastNewton polygon into
red squared ........... coefficients of kappa=Ai
blue diamonds ...... coefficients of lambda=Ai'
newt4a := mynewt(posFk,m,n):
newt4b := mynewt(posFl,m,n):
P4a := pointplot(newt4a,labels=["m deg", "n deg"], symbolsize=25, symbol=box,color=red):
P4b := pointplot(newt4b,labels=["m deg", "n deg"], symbolsize=25, symbol=diamond, color=blue):
P1dom3s := plot(1-m,m=4..5,color=black):
display(P4a,P4b,P1dom1,P1dom2,P1dom3a,myview,myoptionsLo,LegendSize);
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red extremes of Newton polygon
mnmaxR := getMaxNewt(Mord,newt4a):
seq([i,mnmaxR[i]],i=0..Mord);
Nig3JCIiISMhIigiIic3JCIiIiMhIiYiIiQ3JCIiIyEiIzckRiwjISIpRiw3JCIiJSEiJDckIiImISIl
These are the specific values at these points;
we see that we still have some degree in freedom: d and p[4]
for i from 0 to Mord do
i,factor(getel(posFk,i,mnmaxR[i]));
end;
NiQiIiEsJCooIiIlIiIiSSJkRzYiRicpSSJuR0YpIyIiKCIiJyEiIkYv
NiQiIiIsJCosIyIiKSIiJEYjKUkibkc2IiMiIiZGKCEiIkkjYTFHRitGIykiIiMjRjFGKEYjSSJtR0YrRiNGLg==
NiQiIiMsJCooIyIkayIiIioiIiIpSSJuRzYiRiMhIiIpSSJtR0YsRiNGKUYt
NiQiIiQsJCosIyIjO0YjIiIiKUkibkc2IiMiIilGIyEiIikiIiMjRjBGI0YoSSNhMUdGK0YoKUkibUdGK0YjRihGLg==
NiQiIiUsJCooIyIkTyIiIioiIiIpSSJuRzYiIiIkISIiKUkibUdGLEYjRilGLg==
NiQiIiYsJCooIyIkWyMiJE4iIiIiKUkibkc2IiIiJSEiIilJIm1HRixGI0YpRi4=
These are the slopes of the convex hull where the corners are given by the second sequence;
hence, in order to be positive when the slope > -1, we need to choose d>0, e.g. d=1;
note that p[4] is not important here, as the slope first slope -5/12 is less then -7/18;
and in the later regimes it will be dominated by the blue points.
ls,li := getslopes(mnmaxR,Mord);
NiQ3JSMhIiYiIzcjISIiIiIjRig3JiIiIUYpIiIlIiIm
colors := [green,black,brown,blue,olive,red]:
styles := [spacedot,solid , dash , dot , dashdot , longdash , spacedash]:
Draw the conveux hull in red
for i from 1 to nops(ls) do
ls[i];li[i];
tt[i] := plot((mnmaxR[li[i]]-ls[i]*li[i])+ls[i]*m,m=li[i]..li[i+1],color=colors[i mod nops(colors)+1],linestyle=styles[i mod nops(styles)+1],legend=[typeset("Slope: ", ls[i])],legendstyle=[location=right]):
end:
Pconvred := seq(tt[i],i=1..nops(ls)):
#display(%);
display(Pconvred,P4a,myview,myoptionsLo,LegendSize);
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HELVETICAG%(DEFAULTG"#=-%-_LEGENDSTYLEG6#-%*_LOCATIONG6#%'_RIGHTG-%'SYMBOLG6$%(DIAMONDG"#D-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$"$!e!""-%)BOUNDS_YG6#$"$g"!""-%-BOUNDS_WIDTHG6#$"%+S!""-%.BOUNDS_HEIGHTG6#$"%+U!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+g!""-%.BOUNDS_HEIGHTG6#$"%+X!""-%)CHILDRENG6"NiI=
We continue with the blue diamonds, i.e., the coefficients of Ai'
mnmaxB := getMaxNewt(Mord,newt4b):
seq([i,mnmaxB[i]],i=0..Mord);
Nig3JCIiISMhIiUiIiQ3JCIiIiEiIzckIiIjIyEiKEYnNyRGJ0YtNyQiIiUjISM1Ric3JCIiJiMhIzhGJw==
for i from 0 to Mord do
i,factor(getel(posFl,i,mnmaxB[i]));
end;
NiQiIiEqKCkiIiMjIiIiIiIkRigsJiomIiIpRigmSSJxRzYiNiNGKEYoRihGKSEiIkYoKUkibkdGLyMiIiVGKUYx
NiQiIiIsJCoqIyIjSyIiKkYjKUkibkc2IiIiIyEiIkkjYTFHRitGI0kibUdGK0YjRi0=
NiQiIiMsJCosI0YjIiIqIiIiKUYjI0YoIiIkRigpSSJtRzYiRiNGKCwmKiYiI1tGKCZJInFHRi42I0YoRihGKCIjbCEiIkYoKUkibkdGLiMiIihGK0Y2Rig=
NiQiIiQsJCoqIyIjayIiKiIiIilJIm5HNiIjIiIoRiMhIiIpIiIjI0YpRiNGKSlJIm1HRixGI0YpRi8=
NiQiIiUsJCoqIyIjUyIiKiIiIilJIm5HNiIjIiM1IiIkISIiKSIiIyNGKUYvRikpSSJtR0YsRiNGKUYw
NiQiIiYsJCoqIyIkNygiJE4iIiIiKUkibkc2IiMiIzgiIiQhIiIpIiIjI0YpRi9GKSlJIm1HRixGI0YpRjA=
Again we derive the slopes of the convex hull and its corners;
note that if we choose q[1]=3/8 we eliminate the first term and decrease the slope.
TODO:
This will be useful in the proof of Lemma 5.3, in the same way as it was used in the proof of Lemma 4.4.
ls,li := getslopes(mnmaxB,Mord);
NiQ3JCMhIiIiIiRGJTclIiIhRiYiIiY=
colors := [green,olive,blue,black,brown,blue,red]:
styles := [spacedot,solid , dot , dash , dashdot , longdash , spacedash]:
Draw the conveux hull in blue which still includes the term of order Theta(n^(-4/3))
for i from 1 to nops(ls) do
ls[i];li[i];
tt[i] := plot((mnmaxB[li[i]]-ls[i]*li[i])+ls[i]*m,m=li[i]..li[i+1],color=colors[i mod nops(colors)+1],linestyle=styles[i mod nops(styles)+1],legend=[typeset("Slope: ", ls[i])],legendstyle=[location=right]):
end:
Pconvblue := seq(tt[i],i=1..nops(ls)):
#display(%);
display(Pconvblue,P4b,myview,myoptionsLo,LegendSize);
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We kill this term by setting q[1] = 3/8 and recompute the Newton polygons.
In the next picture the left-top blue point disappeared.
(note that the coefficients of lower order terms of posFk change as well, which is why we recompute them; however the picture will not change as the convex hull is not influenced by this change.)
posFk2 := coeff(subs(q[1]=3/8,posFfabcde),kappa):indets(%);
newt4a2 := mynewt(posFk2,m,n):
P4a2 := pointplot(newt4a2,labels=["m deg", "n deg"], symbolsize=25, symbol=box, color=red):
posFl2 := coeff(subs(q[1]=3/8,posFfabcde),lambda):indets(%);
newt4b2 := mynewt(posFl2,m,n):
P4b2 := pointplot(newt4b2,labels=["m deg", "n deg"], symbolsize=25, symbol=diamond, color=blue):
display(P4a2,P4b2,P1dom1,P1dom2,P1dom3a,myview,myoptionsLo,LegendSize);
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Recompute blue with q[1]=1/4
mnmaxB := getMaxNewt(Mord,newt4b2):
seq([i,mnmaxB[i]],i=0..Mord);
Nig3JCIiISMhIiQiIiM3JCIiIiEiIzckRicjISIoIiIkNyRGLkYsNyQiIiUjISM1Ri43JCIiJiMhIzhGLg==
for i from 0 to Mord do
i,factor(getel(posFl2,i,mnmaxB[i]));
end;
NiQiIiEsJCooIiIlIiIiKUkibkc2IiMiIiQiIiMhIiIpRi0jRidGLEYnRic=
NiQiIiIsJCoqIyIjSyIiKkYjKUkibkc2IiIiIyEiIkkjYTFHRitGI0kibUdGK0YjRi0=
NiQiIiMsJCoqIyIjJSoiIioiIiIpSSJuRzYiIyIiKCIiJCEiIilGIyNGKUYvRikpSSJtR0YsRiNGKUYw
NiQiIiQsJCoqIyIjayIiKiIiIilJIm5HNiIjIiIoRiMhIiIpIiIjI0YpRiNGKSlJIm1HRixGI0YpRi8=
NiQiIiUsJCoqIyIjUyIiKiIiIilJIm5HNiIjIiM1IiIkISIiKSIiIyNGKUYvRikpSSJtR0YsRiNGKUYw
NiQiIiYsJCoqIyIkNygiJE4iIiIiKUkibkc2IiMiIzgiIiQhIiIpIiIjI0YpRi9GKSlJIm1HRixGI0YpRjA=
And we get new slopes, yet at the same m powers given in the second sequence;
here we need that the term m^3 is negative as it will dominate in the regime when Ai' is negative,
TODO:
therefore we have to choose p[4]>2/9 (note that this also makes m^5 and m^7 multiplied by Ai' positive)
ls,li := getslopes(mnmaxB,Mord);
NiQ3JCMhIiYiIz0hIiI3JSIiISIiJCIiJg==
Draw the new conveux hull in blue
for i from 1 to nops(ls) do
ls[i];li[i];
tt[i] := plot((mnmaxB[li[i]]-ls[i]*li[i])+ls[i]*m,m=li[i]..li[i+1],color=colors[i mod nops(colors)+1],linestyle=styles[i mod nops(styles)+1],legend=[typeset("Slope: ", ls[i])],legendstyle=[location=right]):
end:
Pconvblue := seq(tt[i],i=1..nops(ls)):
#display(%);
Plot the difference to before:
Only the solid circle on the top-left disappeared.
P4bdiffshort := pointplot([0,-4/3],labels=["m deg", "n deg"], symbolsize=25, symbol=solidcircle, color=blue):
display(Pconvblue,P4bdiffshort,P4b2,myview,myoptionsLo,LegendSize);
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Minimal DFAs recoginizing a finite binary language - Upper bound
Xhat
This is Y hat
Xansatz := (n,m) -> (1-2*m^2/(3*n)+3*m/(8*n)+1/3*m^4/n^2)* AiryAi(a1+2^(1/3)*(m+1)/n^(1/3));
Zio2JEkibkc2IkkibUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYsKiIiIkYsKigjIiIjIiIkRiwpRiZGL0YsRiQhIiJGMiooI0YwIiIpRixGJkYsRiRGMkYsKigjRixGMEYsKUYmIiIlRiwpRiRGL0YyRixGLC1JJ0FpcnlBaUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCZJI2ExR0YlRiwqKClGL0Y3RiwsJkYmRixGLEYsRiwpRiRGN0YyRixGLEYlRiVGJQ==
Sansatz := n -> 2 + c*n^(-2/3) + pterm/n + 1/(n^(7/6));
Zio2I0kibkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCoiIiMiIiIqJkkiY0dGJUYrKUYkI0YqIiIkISIiRisqJkkmcHRlcm1HRiVGK0YkRjFGKyokKUYkIyIiKCIiJ0YxRitGJUYlRiU=
lower bound
(only difference to upper bound is missing factor (n-m-4)/(n-m-2) multiplied with the last two terms;
as it is a lower bound, and we still want to prove positivity, we multiply the full equation with -1)
posansatz := -(-XX(n,m)*SS(n)*SS(n-1)*SS(n-2)
+ (n-m+2)/(n+m)*XX(n-1,m-1)*SS(n-1)*SS(n-2)
+ (n-m-1)/(n-m)*XX(n-1,m+1)*SS(n-1)*SS(n-2)
+ 1/(n-m)*XX(n-2,m+2)*SS(n-2)
+ 1/(n+m)*XX(n-3,m+1)
);
LCwqKi1JI1hYRzYiNiRJIm5HRiZJIm1HRiYiIiItSSNTU0dGJjYjRihGKi1GLDYjLCZGKEYqRiohIiJGKi1GLDYjLCZGKEYqIiIjRjFGKkYqKiwsKEYoRipGKUYxRjVGKkYqLCZGKEYqRilGKkYxLUYlNiRGMCwmRilGKkYqRjFGKkYuRipGMkYqRjEqLCwoRihGKkYpRjFGKkYxRiosJkYoRipGKUYxRjEtRiU2JEYwLCZGKUYqRipGKkYqRi5GKkYyRipGMSooRj5GMS1GJTYkRjQsJkYpRipGNUYqRipGMkYqRjEqJkY4RjEtRiU2JCwmRihGKiIiJEYxRkFGKkYx
posXS := map(simplify,subs(XX=Xansatz,SS=Sansatz,posansatz)):
For a large n this function of m seems to be positive
Digits:=20:
e1 := subs(csubs,pterm=29/12,a1=A1,posXS):
N := 100000;
M := floor(N^(1/2));
P1 := plot([seq([mm,(subs(n=N,m=mm,e1))],mm=0..M)]):display(P1);
IicrKzU=
IiQ7JA==
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Prove it
We start with the ansatz of Yhat in Lemma 5.3.
Recall the general ansatz
facAiryUp*Airy(a1+2^(1/3)*(m+1)/n^(1/3));;
SF(n);
KiYsKCIiIkYkKiYsLComKUkibUc2IiIiJUYkJkkicEdGKjYjRitGJEYkKiYpRikiIiRGJCZGLTYjRjFGJEYkKiYpRikiIiNGJCZGLTYjRjZGJEYkKiZGKUYkJkYtNiNGJEYkRiQmRi02IyIiIUYkRiQpSSJuR0YqRjYhIiJGJComLCgqJkY1RiQmSSJxR0YqRjhGJEYkKiZGKUYkJkZGRjtGJEYkJkZGRj1GJEYkRkBGQUYkRiQtSSVBaXJ5R0YqNiMsJkkjYTFHRipGJCooKUY2I0YkRjFGJCwmRilGJEYkRiRGJClGQEZRRkFGJEYk
LCxJImFHNiIiIiIqJkkiYkdGJEYlKUkibkdGJCNGJSIiJCEiIkYlKiZJImNHRiRGJSlGKSMiIiNGK0YsRiUqJkkmcHRlcm1HRiRGJUYpRixGJSomSSJkR0YkRiUpRikjIiIoIiInRixGJQ==
Substitute ansatz into sequence we want to be positive for large n and all m
posF := map(expand, subs(XX=XFU,SS=SF,posansatz)):indets(%);
PEJJImFHNiJJI2ExR0YkSSJiR0YkSSJjR0YkSSJkR0YkSSZrYXBwYUdGJEknbGFtYmRhR0YkSSJtR0YkSSJuR0YkSSZwdGVybUdGJCZJInBHRiQ2IyIiISZGLzYjIiIiJkYvNiMiIiMmRi82IyIiJCZGLzYjIiIlJkkicUdGJEYwJkY/RjMmRj9GNiokKUYsIyIiKUY6ISIiKiQpRiwjIiIoRjpGRiokKUYsI0ZKIiInRkYqJClGLCMiIiZGOkZGKiQpRiwjRj1GOkZGKiQpRiwjRjdGOkZGKiQpRiwjRjRGOkZGKiQpLCZGLEY0RjpGRkZlbkZGKiQpLCZGLEY0RjdGRkZNRkYqJClGW29GWEZGKiQpRltvRmVuRkYqJCksJkYsRjRGNEZGRk1GRiokKUZib0ZYRkYqJClGYm9GZW5GRg==
The error terms are (to check, look at posFc)
simplify(O((2^(1/3)*(m+1)/n^(1/3)-2^(1/3)*m/n^(1/3))^ordAiUp));
simplify(O((2^(1/3)*(m)/(n-1)^(1/3)-2^(1/3)*m/n^(1/3))^ordAiUp));
simplify(O((2^(1/3)*(m+2)/(n-1)^(1/3)-2^(1/3)*m/n^(1/3))^ordAiUp));
LUkiT0clKnByb3RlY3RlZEc2IywkKigiI2siIiIpIiIjI0YpIiIkRikpSSJuRzYiIyIjPkYtISIiRik=
LUkiT0clKnByb3RlY3RlZEc2IywkKi4iI2siIiIpIiIjI0YpIiIkRikpSSJtRzYiIiM+RikpLCYqJCksJkkibkdGMEYpRikhIiJGLEYpRikqJClGN0YsRilGOEYxRikpRjYjRjFGLUY4KUY3RjxGOEY4
LUkiT0clKnByb3RlY3RlZEc2IywkKiwiI2siIiIpIiIjI0YpIiIkRikpLCgqJkkibUc2IkYpKSwmSSJuR0YyRilGKSEiIkYsRilGKSomKUY1RixGKUYxRilGNiomRitGKUY4RilGNiIjPkYpKUY0I0Y6Ri1GNilGNUY8RjZGNg==
remove error terms
posFd := convert(posF,polynom):
Next we fit the display size of the plots in the n (Nord) and m (Mord) direction.
(Note that everything up to ordAi is computed, but possibly not shown)
Nord := -ordAiUp/3;
Mord := floor(ordAiUp/3)+1;
myview := view=[0..Mord,Nord..0]:
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Expand again with respect to n,
these are then our unknowns
posFe := series(posFd,n=infinity,ceil(-Nord)+1):indets(%);
posFf := convert(%%,polynom):
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
The mynewt function computes the Newton polygon of posFf
newt1 := mynewt(posFf,m,n):
First Newton polygon, where no unknowns have been fixed
P1 := pointplot(newt1,myoptionsUp,color=blue):
display(P1,myview);
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Here, we want to kill the element (0,0)
getel(posFf,0,0);
LCYqJilJImFHNiIiIiQiIiJJJmthcHBhR0YmRihGKCooIiIjRigpRiVGK0YoRilGKCEiIg==
Set a=2
posFfa := expand(simplify(subs(a=2,posFf))) assuming n::posint,m::posint:
newta := mynewt(posFfa,m,n):
All blue points have been eliminated, and only the red ones remain
P1a := pointplot(newta,myoptionsUp,color=red):
display(P1,P1a,myview);
6,-%'POINTSG6]x7$$""!!""$!#g!""7$$""!!""$!#]!""7$$""!!""$!#S!""7$$""!!""$!#I!""7$$""!!""$!#?!""7$$""!!""$!#5!""7$$""!!""$""!!""7$$""!!""$!1LLLLLLLo!#:7$$""!!""$!1nmmmmmmh!#:7$$""!!""$!1LLLLLLLe!#:7$$""!!""$!1nmmmmmm^!#:7$$""!!""$!1LLLLLLL[!#:7$$""!!""$!1nmmmmmmT!#:7$$""!!""$!2NLLLLLL$Q!#;7$$""!!""$!1nmmmmmmm!#:7$$""!!""$!1LLLLLLLj!#:7$$""!!""$!2lmmmmmm;$!#;7$$""!!""$!1nmmmmmmc!#:7$$""!!""$!2NLLLLLL$G!#;7$$""!!""$!1LLLLLLL`!#:7$$""!!""$!1nmmmmmmY!#:7$$""!!""$!#l!""7$$""!!""$!1LLLLLLLV!#:7$$""!!""$!2lmmmmmm;#!#;7$$""!!""$!#b!""7$$""!!""$!2lmmmmmmm$!#;7$$""!!""$!2LLLLLLL$=!#;7$$""!!""$!2NLLLLLLL$!#;7$$""!!""$!#X!""7$$""!!""$!2lmmmmmmm#!#;7$$""!!""$!#N!""7$$""!!""$!2NLLLLLLL#!#;7$$""!!""$!2nmmmmmm;"!#;7$$""!!""$!#D!""7$$""!!""$!2nmmmmmmm"!#;7$$""!!""$!2LLLLLLLL"!#;7$$""!!""$!#:!""7$$""!!""$!1mmmmmmmm!#;7$$""!!""$!1LLLLLLLL!#;7$$"#5!""$!#g!""7$$"#5!""$!#]!""7$$"#5!""$!#S!""7$$"#5!""$!#I!""7$$"#5!""$!#?!""7$$"#5!""$!#5!""7$$"#5!""$!1LLLLLLLo!#:7$$"#5!""$!1nmmmmmmh!#:7$$"#5!""$!1LLLLLLLe!#:7$$"#5!""$!1nmmmmmm^!#:7$$"#5!""$!1LLLLLLL[!#:7$$"#5!""$!1nmmmmmmT!#:7$$"#5!""$!2NLLLLLL$Q!#;7$$"#5!""$!1nmmmmmmm!#:7$$"#5!""$!1LLLLLLLj!#:7$$"#5!""$!2lmmmmmm;$!#;7$$"#5!""$!1nmmmmmmc!#:7$$"#5!""$!2NLLLLLL$G!#;7$$"#5!""$!1LLLLLLL`!#:7$$"#5!""$!1nmmmmmmY!#:7$$"#5!""$!#l!""7$$"#5!""$!1LLLLLLLV!#:7$$"#5!""$!2lmmmmmm;#!#;7$$"#5!""$!#b!""7$$"#5!""$!2lmmmmmmm$!#;7$$"#5!""$!2NLLLLLLL$!#;7$$"#5!""$!#X!""7$$"#5!""$!2lmmmmmmm#!#;7$$"#5!""$!#N!""7$$"#5!""$!2NLLLLLLL#!#;7$$"#5!""$!#D!""7$$"#5!""$!2nmmmmmmm"!#;7$$"#5!""$!2LLLLLLLL"!#;7$$"#?!""$!#g!""7$$"#?!""$!#]!""7$$"#?!""$!#S!""7$$"#?!""$!#I!""7$$"#?!""$!#?!""7$$"#?!""$!#5!""7$$"#?!""$!1LLLLLLLo!#:7$$"#?!""$!1nmmmmmmh!#:7$$"#?!""$!1LLLLLLLe!#:7$$"#?!""$!1nmmmmmm^!#:7$$"#?!""$!1LLLLLLL[!#:7$$"#?!""$!1nmmmmmmT!#:7$$"#?!""$!2NLLLLLL$Q!#;7$$"#?!""$!1nmmmmmmm!#:7$$"#?!""$!1LLLLLLLj!#:7$$"#?!""$!2lmmmmmm;$!#;7$$"#?!""$!1nmmmmmmc!#:7$$"#?!""$!2NLLLLLL$G!#;7$$"#?!""$!1LLLLLLL`!#:7$$"#?!""$!1nmmmmmmY!#:7$$"#?!""$!#l!""7$$"#?!""$!1LLLLLLLV!#:7$$"#?!""$!2lmmmmmm;#!#;7$$"#?!""$!#b!""7$$"#?!""$!2lmmmmmmm$!#;7$$"#?!""$!2NLLLLLLL$!#;7$$"#?!""$!#X!""7$$"#?!""$!2lmmmmmmm#!#;7$$"#?!""$!#N!""7$$"#?!""$!2NLLLLLLL#!#;7$$"#?!""$!#D!""7$$"#?!""$!2nmmmmmmm"!#;7$$"#?!""$!2LLLLLLLL"!#;7$$"#I!""$!#g!""7$$"#I!""$!#]!""7$$"#I!""$!#S!""7$$"#I!""$!#I!""7$$"#I!""$!#?!""7$$"#I!""$!1LLLLLLLo!#:7$$"#I!""$!1nmmmmmmh!#:7$$"#I!""$!1LLLLLLLe!#:7$$"#I!""$!1nmmmmmm^!#:7$$"#I!""$!1LLLLLLL[!#:7$$"#I!""$!1nmmmmmmT!#:7$$"#I!""$!2NLLLLLL$Q!#;7$$"#I!""$!1nmmmmmmm!#:7$$"#I!""$!1LLLLLLLj!#:7$$"#I!""$!2lmmmmmm;$!#;7$$"#I!""$!1nmmmmmmc!#:7$$"#I!""$!1LLLLLLL`!#:7$$"#I!""$!1nmmmmmmY!#:7$$"#I!""$!#l!""7$$"#I!""$!1LLLLLLLV!#:7$$"#I!""$!#b!""7$$"#I!""$!2lmmmmmmm$!#;7$$"#I!""$!2NLLLLLLL$!#;7$$"#I!""$!#X!""7$$"#I!""$!2lmmmmmmm#!#;7$$"#I!""$!#N!""7$$"#I!""$!2NLLLLLLL#!#;7$$"#S!""$!#g!""7$$"#S!""$!#]!""7$$"#S!""$!#S!""7$$"#S!""$!#I!""7$$"#S!""$!#?!""7$$"#S!""$!1LLLLLLLo!#:7$$"#S!""$!1nmmmmmmh!#:7$$"#S!""$!1LLLLLLLe!#:7$$"#S!""$!1nmmmmmm^!#:7$$"#S!""$!1LLLLLLL[!#:7$$"#S!""$!1nmmmmmmT!#:7$$"#S!""$!2NLLLLLL$Q!#;7$$"#S!""$!1nmmmmmmm!#:7$$"#S!""$!1LLLLLLLj!#:7$$"#S!""$!2lmmmmmm;$!#;7$$"#S!""$!1nmmmmmmc!#:7$$"#S!""$!1LLLLLLL`!#:7$$"#S!""$!1nmmmmmmY!#:7$$"#S!""$!#l!""7$$"#S!""$!1LLLLLLLV!#:7$$"#S!""$!#b!""7$$"#S!""$!2lmmmmmmm$!#;7$$"#S!""$!2NLLLLLLL$!#;7$$"#S!""$!#X!""7$$"#S!""$!2lmmmmmmm#!#;7$$"#S!""$!#N!""7$$"#S!""$!2NLLLLLLL#!#;7$$"#]!""$!#g!""7$$"#]!""$!#]!""7$$"#]!""$!#S!""7$$"#]!""$!#I!""7$$"#]!""$!1LLLLLLLo!#:7$$"#]!""$!1nmmmmmmh!#:7$$"#]!""$!1LLLLLLLe!#:7$$"#]!""$!1nmmmmmm^!#:7$$"#]!""$!1LLLLLLL[!#:7$$"#]!""$!1nmmmmmmT!#:7$$"#]!""$!1nmmmmmmm!#:7$$"#]!""$!1LLLLLLLj!#:7$$"#]!""$!1nmmmmmmc!#:7$$"#]!""$!1LLLLLLL`!#:7$$"#]!""$!1nmmmmmmY!#:7$$"#]!""$!#l!""7$$"#]!""$!1LLLLLLLV!#:7$$"#]!""$!#b!""7$$"#]!""$!2lmmmmmmm$!#;7$$"#]!""$!2NLLLLLLL$!#;7$$"#]!""$!#X!""7$$"#g!""$!#g!""7$$"#g!""$!#]!""7$$"#g!""$!#S!""7$$"#g!""$!1LLLLLLLo!#:7$$"#g!""$!1nmmmmmmh!#:7$$"#g!""$!1LLLLLLLe!#:7$$"#g!""$!1nmmmmmm^!#:7$$"#g!""$!1nmmmmmmm!#:7$$"#g!""$!1LLLLLLLj!#:7$$"#g!""$!1nmmmmmmc!#:7$$"#g!""$!1LLLLLLL`!#:7$$"#g!""$!1nmmmmmmY!#:7$$"#g!""$!#l!""7$$"#g!""$!1LLLLLLLV!#:7$$"#g!""$!#b!""7$$"#q!""$!#g!""7$$"#q!""$!#]!""7$$"#q!""$!1LLLLLLLo!#:7$$"#q!""$!1nmmmmmmh!#:7$$"#q!""$!1nmmmmmmm!#:7$$"#q!""$!1LLLLLLLj!#:7$$"#q!""$!1nmmmmmmc!#:7$$"#q!""$!1LLLLLLL`!#:7$$"#q!""$!#l!""7$$"#!)!""$!#g!""7$$"#!)!""$!1nmmmmmmm!#:7$$"#!)!""$!1LLLLLLLj!#:-%&COLORG6&%$RGBG$""!!""$""!!""$"#5!""-%'SYMBOLG6$%(DIAMONDG"#?-%'POINTSG6gw7$$""!!""$!#g!""7$$""!!""$!#]!""7$$""!!""$!#S!""7$$""!!""$!#I!""7$$""!!""$!#?!""7$$""!!""$!#5!""7$$""!!""$!1LLLLLLLo!#:7$$""!!""$!1nmmmmmmh!#:7$$""!!""$!1LLLLLLLe!#:7$$""!!""$!1nmmmmmm^!#:7$$""!!""$!1LLLLLLL[!#:7$$""!!""$!1nmmmmmmT!#:7$$""!!""$!2NLLLLLL$Q!#;7$$""!!""$!1nmmmmmmm!#:7$$""!!""$!1LLLLLLLj!#:7$$""!!""$!2lmmmmmm;$!#;7$$""!!""$!1nmmmmmmc!#:7$$""!!""$!2NLLLLLL$G!#;7$$""!!""$!1LLLLLLL`!#:7$$""!!""$!1nmmmmmmY!#:7$$""!!""$!#l!""7$$""!!""$!1LLLLLLLV!#:7$$""!!""$!2lmmmmmm;#!#;7$$""!!""$!#b!""7$$""!!""$!2lmmmmmmm$!#;7$$""!!""$!2LLLLLLL$=!#;7$$""!!""$!2NLLLLLLL$!#;7$$""!!""$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b=0 is forced due to the term n^{-1/3)
getel(posFfa,0,-1/3);
#getel(posFfa,2,-4/3);
LCQqKiIiJSIiIilJIm5HNiIjRiUiIiQhIiJJJmthcHBhR0YoRiVJImJHRihGJUYl
set a=2, b=0
posFfab := expand(simplify(subs(b=0,posFfa))) assuming n::posint,m::posint:
newtab := mynewt(posFfab,m,n):
Now only the green points remain.
P1ab := pointplot(newtab,myoptionsUp,color=green):
display(P1,P1a,P1ab,myview);
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at this point we find our choice for c, which we heuristically computed already before in Section 3
csubs;
factor(getel(posFfab,0,-2/3));factor(subs(csubs,%));
factor(getel(posFfab,1,-4/3));factor(subs(csubs,%));
factor(getel(posFfab,2,-5/3));factor(subs(csubs,%));
L0kiY0c2IiomSSNhMUdGJCIiIikiIiMjRikiIiRGJw==
LCQqKiIiJSIiIkkma2FwcGFHNiJGJSwmKiZJI2ExR0YnRiUpIiIjI0YsIiIkRiVGJUkiY0dGJyEiIkYlKUkibkdGJ0YtRjBGMA==
IiIh
LCQqLiMiIzsiIiQiIiIpIiIjI0YnRiZGJ0knbGFtYmRhRzYiRidJIm1HRixGJywmKiZGJkYnJkkicUdGLDYjRilGJ0YnRilGJ0YnKUkibkdGLCMiIiVGJiEiIkY3
LCQqLiMiIzsiIiQiIiIpIiIjI0YnRiZGJ0knbGFtYmRhRzYiRidJIm1HRixGJywmKiZGJkYnJkkicUdGLDYjRilGJ0YnRilGJ0YnKUkibkdGLCMiIiVGJiEiIkY3
LCQqLiIiJSIiIkkma2FwcGFHNiJGJSlJIm1HRiciIiNGJSZJInFHRic2I0YqRiUsJiomSSNhMUdGJ0YlKUYqI0YqIiIkRiVGJUkiY0dGJyEiIkYlKUkibkdGJyMiIiZGM0Y1RjU=
IiIh
set a=2, b=0, c=a1*2^(2/3)
posFfabc := expand(simplify(subs(csubs,posFfab))) assuming n::posint,m::posint:
newtabc := mynewt(posFfabc,m,n):
P1abc := pointplot(newtabc,myoptionsUp,color=yellow):
display(P1,P1a,P1ab,P1abc,myview);
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Here we get q[2] and pterm
factor(getel(posFfabc,0,-1));
factor(getel(posFfabc,1,-4/3));
solve({%%,%},{q[2],pterm});
KihJJmthcHBhRzYiIiIiLCgqJiIiJUYlSSZwdGVybUdGJEYlRiUqJiIiKUYlJkkicUdGJDYjIiIjRiUhIiIiIzpGMEYlSSJuR0YkRjA=
LCQqLiMiIzsiIiQiIiIpIiIjI0YnRiZGJ0knbGFtYmRhRzYiRidJIm1HRixGJywmKiZGJkYnJkkicUdGLDYjRilGJ0YnRilGJ0YnKUkibkdGLCMiIiVGJiEiIkY3
PCQvSSZwdGVybUc2IiMiI0giIzcvJkkicUdGJTYjIiIjIyEiIyIiJA==
set a=2, b=0,c,pterm=29/12,q[2]=-2/3
posFfabcd := expand(simplify(subs(pterm=29/12,q[2]=-2/3,posFfabc))) assuming n::posint,m::posint:
newtabcd := mynewt(posFfabcd,m,n):
only the brown points remain
Now all points are strictly below n^{-1}
P1abcd := pointplot(newtabcd,myoptionsUp,color=brown):
display(P1,P1a,P1ab,P1abc,P1abcd,myview);
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Here are the dominating corners and we see that we have to choose d=1 to have a positive term;
note that we will see that the second term should be negative, as lambda=Ai' is negative for large m,
hence here we will choose a p[4]>2/9 (see below in the decomposition into lambda and kappa contributions)
getel(posFfabcd,0,-7/6);
factor(getel(posFfabcd,3,-14/6));
(14/6-7/6)/3; #slope
LCQqKiIiJSIiIilJIm5HNiIjIiIoIiInISIiSSZrYXBwYUdGKEYlSSJkR0YoRiVGJQ==
LCQqLiMiI0siIioiIiIpIiIjI0YnIiIkRidJJ2xhbWJkYUc2IkYnKUkibUdGLUYrRicsJkYpISIiKiZGJkYnJkkicEdGLTYjIiIlRidGJ0YnKUkibkdGLSMiIihGK0YxRjE=
IyIiKCIjPQ==
and continuing
getel(posFfabcd,5,-20/6);
(20/6-14/6)/2; #slope
LCQqLiMiI0siIiQiIiIpSSJuRzYiIyIjNUYmISIiKSIiIyNGJ0YmRidJJ2xhbWJkYUdGKkYnKUkibUdGKiIiJkYnJkkicEdGKjYjIiIlRidGLQ==
IyIiIiIiIw==
set a=2, b=0,c,pterm=29/12,q[2]=-2/3 and d=1
posFfabcde := expand(simplify(subs(d=1,posFfabcd))) assuming n::posint,m::posint:
newtabcde := mynewt(posFfabcde,m,n):
This is the final result, where only the solid diamonds are non-zero
P1abcde := pointplot(newtabcde,myoptionsUp,symbol=soliddiamond,color=gray):
display(P1,P1a,P1ab,P1abc,P1abcd,P1abcde,myview);
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Plot the boundary and the slopes of the Newton polygon;
Note that we have already proved that there are now points above the blue dotted line
LegendSize := size=[600,450]:
P1dom1 := plot(-7/6-(7/18)*m,m=0..3,color=black,legend=[typeset("Slope: ", -7/18)],legendstyle=[location=right]):
P1dom2 := plot(-5/6-(1/2)*m,m=3..5,color=red,legend=[typeset("Slope: ", -1/2)],legendstyle=[location=right]):
P1dom2b := plot(0-(2/3)*m,m=5..6,color=green,legend=[typeset("Slope: ", -2/3)],legendstyle=[location=right]):
P1dom3a := plot(2-m,m=0..7,color=blue,linestyle=dot,legend=[typeset("Slope: ", -1)],legendstyle=[location=right]):
P1all := display(P1,P1a,P1ab,P1abc,P1abcd,P1abcde,P1dom1,P1dom2,P1dom2b,P1dom3a,myview,LegendSize):
for i from 1 to 10 do
P1dom3[i] := plot([[2+i/2,0],[2+i/2,-i/2]],color=blue,linestyle=dot):
end:
display(P1all,seq(P1dom3[i],i=1..10));
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This is the choice for SF
SF(n);
subs(a=2,b=0,csubs,pterm=29/12,d=1,%);
LCxJImFHNiIiIiIqJkkiYkdGJEYlKUkibkdGJCNGJSIiJCEiIkYlKiZJImNHRiRGJSlGKSMiIiNGK0YsRiUqJkkmcHRlcm1HRiRGJUYpRixGJSomSSJkR0YkRiUpRikjIiIoIiInRixGJQ==
LCoiIiMiIiIqKEkjYTFHNiJGJClGIyNGIyIiJEYkKUkibkdGJ0YpISIiRiQqJiMiI0giIzdGJEYsRi1GJCokKUYsIyIiKCIiJ0YtRiQ=
recall
kaplam;
NiQvLUknQWlyeUFpRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJi1JLEFpcnlBaVplcm9zR0YmNiMiIiJGLyooKSIiIyNGLyIiJEYvSSJtR0YpRi8pSSJuR0YpRjMhIiJGL0kma2FwcGFHRikvLUYlNiRGL0YrSSdsYW1iZGFHRik=
Look at the corners
Here we see the necessary choice: p[4]>2/9 (note that for m>n^(1/3) lambda is negative)
getel(posFfabcde,0,-7/6);
simplify(getel(posFfabcde,3,-14/6));
getel(posFfabcde,5,-20/6);
getel(posFfabcde,6,-4);
getel(posFfabcde,7,-5);
LCQqKCIiJSIiIilJIm5HNiIjIiIoIiInISIiSSZrYXBwYUdGKEYlRiU=
LCQqLiMiI0siIioiIiIpIiIjI0YnIiIkRidJJ2xhbWJkYUc2IkYnKUkibUdGLUYrRicsJkYpISIiKiZGJkYnJkkicEdGLTYjIiIlRidGJ0YnKUkibkdGLSMiIihGK0YxRjE=
LCQqLiMiI0siIiQiIiIpSSJuRzYiIyIjNUYmISIiKSIiIyNGJ0YmRidJJ2xhbWJkYUdGKkYnKUkibUdGKiIiJkYnJkkicEdGKjYjIiIlRidGLQ==
LCQqLCMiI28iIiQiIiIpSSJuRzYiIiIlISIiSSZrYXBwYUdGKkYnKUkibUdGKiIiJ0YnJkkicEdGKjYjRitGJ0Ys
LCQqLCMiJEMiIiNYIiIiKUkibkc2IiIiJiEiIkkma2FwcGFHRipGJylJIm1HRioiIihGJyZJInBHRio2IyIiJUYnRiw=
Now split the black dots into the contributions from Ai and Ai'
indets(posFfabcd);
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
Sanity check that there are no other contributions
subs(kappa=0,lambda=0,posFfabcd);
IiIh
Extract the coefficients of kappa=Ai and lambda=Ai'
and treat then seperately
posFk := coeff(posFfabcd,kappa):indets(%);
posFl := coeff(posFfabcd,lambda):indets(%);
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
We color the non-zero nodes of the lastNewton polygon into
red squared ........... coefficients of kappa=Ai
blue diamonds ...... coefficients of lambda=Ai'
newt4a := mynewt(posFk,m,n):
newt4b := mynewt(posFl,m,n):
P4a := pointplot(newt4a,labels=["m deg", "n deg"], symbolsize=15, symbol=box,color=red):
P4b := pointplot(newt4b,labels=["m deg", "n deg"], symbolsize=15, symbol=diamond, color=blue):
P1dom3s := plot(1-m,m=4..5,color=black):
display(P4a,P4b,P1dom1,P1dom2,P1dom2b,P1dom3a,myview,myoptionsUp,LegendSize);
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EFAULTG"#=-%-_LEGENDSTYLEG6#-%*_LOCATIONG6#%'_RIGHTG-%'SYMBOLG6$%(DIAMONDG"#?-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$"$!e!""-%)BOUNDS_YG6#$"$g"!""-%-BOUNDS_WIDTHG6#$"%5S!""-%.BOUNDS_HEIGHTG6#$"%5U!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+g!""-%.BOUNDS_HEIGHTG6#$"%+X!""-%)CHILDRENG6"Ig==
red extremes of Newton polygon
mnmaxR := getMaxNewt(Mord,newt4a):
seq([i,mnmaxR[i]],i=0..Mord);
Nio3JCIiISMhIigiIic3JCIiIiMhIiYiIiQ3JCIiIyEiIzckRiwjISIpRiw3JCIiJSEiJDckIiImIyEjNkYsNyRGJyEiJTckIiIoRis=
These are the specific values at these points;
we see that we still have some degree in freedom: d and p[4]
for i from 0 to Mord do
i,factor(getel(posFk,i,mnmaxR[i]));
end;
NiQiIiEsJCooIiIlIiIiSSJkRzYiRicpSSJuR0YpIyIiKCIiJyEiIkYn
NiQiIiIsJCosIyIiKSIiJEYjKUkibkc2IiMiIiZGKCEiIkkjYTFHRitGIykiIiMjRjFGKEYjSSJtR0YrRiNGIw==
NiQiIiMsJCoqIyIiJSIiKiIiIilJIm1HNiJGI0YpLCYqJiIkMyJGKSZJInBHRiw2I0YnRilGKSIjVCEiIkYpKUkibkdGLEYjRjRGNA==
NiQiIiQsJCouIyIjO0YjIiIiKSIiIyNGKkYjRihJI2ExRzYiRigpSSJtR0YtRiNGKCwmKiYiIidGKCZJInBHRi02IyIiJUYoRihGKCEiIkYoKUkibkdGLSMiIilGI0Y3Rjc=
NiQiIiUsJCoqIyIiKSIiKiIiIilJIm1HNiJGI0YpLCYiIzwhIiIqJiIkSyJGKSZJInBHRiw2I0YjRilGKUYpKUkibkdGLCIiJEYvRi8=
NiQiIiYsJCouIiIpIiIiKUkibkc2IiMiIzYiIiQhIiIpIiIjI0YwRi1GJ0kjYTFHRipGJylJIm1HRipGI0YnJkkicEdGKjYjIiIlRidGLg==
NiQiIicsJCoqIyIjbyIiJCIiIilJIm5HNiIiIiUhIiIpSSJtR0YsRiNGKSZJInBHRiw2I0YtRilGLg==
NiQiIigsJCoqIyIkQyIiI1giIiIpSSJuRzYiIiImISIiKUkibUdGLEYjRikmSSJwR0YsNiMiIiVGKUYu
These are the slopes of the convex hull where the corners are given by the second sequence;
hence, in order to be positive when the slope > -1, we need to choose d>0, e.g. d=1;
note that p[4] is not important here, as the slope first slope -5/12 is less then -7/18;
and in the later regimes it will be dominated by the blue points.
ls,li := getslopes(mnmaxR,Mord);
NiQ3JSMhIiYiIzcjISIiIiIjRig3JiIiIUYpIiInIiIo
colors := [green,black,brown,blue,olive,red]:
styles := [spacedot,solid , dash , dot , dashdot , longdash , spacedash]:
Draw the conveux hull in red
for i from 1 to nops(ls) do
ls[i];li[i];
tt[i] := plot((mnmaxR[li[i]]-ls[i]*li[i])+ls[i]*m,m=li[i]..li[i+1],color=colors[i mod nops(colors)+1],linestyle=styles[i mod nops(styles)+1],legend=[typeset("Slope: ", ls[i])],legendstyle=[location=right]):
end:
Pconvred := seq(tt[i],i=1..nops(ls)):
#display(%);
display(Pconvred,P4a,myview,myoptionsUp,LegendSize);
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We continue with the blue diamonds, i.e., the coefficients of Ai'
mnmaxB := getMaxNewt(Mord,newt4b):
seq([i,mnmaxB[i]],i=0..Mord);
Nio3JCIiISMhIiUiIiQ3JCIiIiEiIzckIiIjIyEiKEYnNyRGJ0YtNyQiIiUjISM1Ric3JCIiJkYyNyQiIicjISM4Ric3JCIiKCMhIztGJw==
for i from 0 to Mord do
i,factor(getel(posFl,i,mnmaxB[i]));
end;
NiQiIiEsJCooKSIiIyMiIiIiIiRGKSwmKiYiIilGKSZJInFHNiI2I0YpRilGKUYqISIiRikpSSJuR0YwIyIiJUYqRjJGMg==
NiQiIiIsJCoqIyIjSyIiKkYjKUkibkc2IiIiIyEiIkkjYTFHRitGI0kibUdGK0YjRiM=
NiQiIiMsJCosI0YjIiIqIiIiKUYjI0YoIiIkRigpSSJtRzYiRiNGKCwqKiYiJDMiRigmSSJwR0YuNiNGK0YoRigqJiIkOyNGKCZGMzYjIiIlRihGKComIiNbRigmSSJxR0YuNiNGKEYoRigiI2whIiJGKClJIm5HRi4jIiIoRitGQEZA
NiQiIiQsJCosIyIjSyIiKiIiIikiIiMjRilGI0YpKUkibUc2IkYjRiksJiomRihGKSZJInBHRi82IyIiJUYpRilGKyEiIkYpKUkibkdGLyMiIihGI0Y2RjY=
NiQiIiUsJCosIyIiIiIiKkYnKSIiIyNGJyIiJEYnKUkibUc2IkYjRicsKComIiMnKkYnJkkicEdGLzYjRixGJ0YnKiYiJFwmRicmRjQ2I0YjRidGJyIjUyEiIkYnKUkibkdGLyMiIzVGLEY7Rjs=
NiQiIiYsJCosIyIjSyIiJCIiIilJIm5HNiIjIiM1RighIiIpIiIjI0YpRihGKSlJIm1HRixGI0YpJkkicEdGLDYjIiIlRilGLw==
NiQiIicsJCosIyIjPyIiJCIiIilJIm5HNiIjIiM4RighIiIpIiIjI0YpRihGKSlJIm1HRixGI0YpJkkicEdGLDYjIiIlRilGLw==
NiQiIigsJCosIyIkYyQiI1giIiIpSSJuRzYiIyIjOyIiJCEiIikiIiMjRilGL0YpKUkibUdGLEYjRikmSSJwR0YsNiMiIiVGKUYw
Again we derive the slopes of the convex hull and its corners;
note that if we choose q[1]=3/8 we eliminate the first term and decrease the slope.
TODO:
This will be useful in the proof of Lemma 5.3, in the same way as it was used in the proof of Lemma 4.4.
ls,li := getslopes(mnmaxB,Mord);
NiQ3JSMhIiIiIiQjRiUiIiNGJTcmIiIhRiYiIiYiIig=
colors := [brown,olive,purple,blue,olive,red,black]:
styles := [spacedot,solid , dash , dot , dashdot , longdash , spacedash]:
Draw the conveux hull in blue which still includes the term of order Theta(n^(-4/3))
for i from 1 to nops(ls) do
ls[i];li[i];
tt[i] := plot((mnmaxB[li[i]]-ls[i]*li[i])+ls[i]*m,m=li[i]..li[i+1],color=colors[i mod nops(colors)+1],linestyle=styles[i mod nops(styles)+1],legend=[typeset("Slope: ", ls[i])],legendstyle=[location=right]):
end:
Pconvblue := seq(tt[i],i=1..nops(ls)):
#display(%);
display(Pconvblue,P4b,myview,myoptionsUp,LegendSize);
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We kill this term by setting q[1] = 3/8 and recompute the Newton polygons.
In the next picture the left-top blue point disappeared.
(note that the coefficients of lower order terms of posFk change as well, which is why we recompute them; however the picture will not change as the convex hull is not influenced by this change.)
posFk2 := coeff(subs(q[1]=3/8,p[3]=0,p[2]=0,p[1]=0,p[0]=0,q[0]=0,posFfabcde),kappa):indets(%):
newt4a2 := mynewt(posFk2,m,n):
P4a2 := pointplot(newt4a2,labels=["m deg", "n deg"], symbolsize=15, symbol=box, color=red):
posFl2 := coeff(subs(q[1]=3/8,p[3]=0,p[2]=0,p[1]=0,p[0]=0,q[0]=0,posFfabcde),lambda):indets(%):
newt4b2 := mynewt(posFl2,m,n):
P4b2 := pointplot(newt4b2,labels=["m deg", "n deg"], symbolsize=15, symbol=diamond, color=blue):
display(P4a2,P4b2,P1dom1,P1dom2,P1dom2b,P1dom3a,myview,myoptionsUp,LegendSize);
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EG6#-%*_LOCATIONG6#%'_RIGHTG-%'SYMBOLG6$%(DIAMONDG"#?-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$"$!e!""-%)BOUNDS_YG6#$"$g"!""-%-BOUNDS_WIDTHG6#$"%5S!""-%.BOUNDS_HEIGHTG6#$"%5U!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+g!""-%.BOUNDS_HEIGHTG6#$"%+X!""-%)CHILDRENG6"Ig==
Recompute blue with q[1]=1/4
mnmaxB := getMaxNewt(Mord,newt4b2):
seq([i,mnmaxB[i]],i=0..Mord);
Nio3JCIiISMhIiQiIiM3JCIiIiEiIzckRicjISIoIiIkNyRGLkYsNyQiIiUjISM1Ri43JCIiJkYyNyQiIicjISM4Ri43JCIiKCMhIztGLg==
for i from 0 to Mord do
i,factor(getel(posFl2,i,mnmaxB[i]));
end;
NiQiIiEsJCooIiIlIiIiKUkibkc2IiMiIiQiIiMhIiIpRi0jRidGLEYnRic=
NiQiIiIsJCoqIyIjSyIiKkYjKUkibkc2IiIiIyEiIkkjYTFHRitGI0kibUdGK0YjRiM=
NiQiIiMsJCosI0YjIiIqIiIiKUYjI0YoIiIkRigpSSJtRzYiRiNGKCwmIiNaISIiKiYiJDsjRigmSSJwR0YuNiMiIiVGKEYoRigpSSJuR0YuIyIiKEYrRjFGMQ==
NiQiIiQsJCosIyIjSyIiKiIiIikiIiMjRilGI0YpKUkibUc2IkYjRiksJiomRihGKSZJInBHRi82IyIiJUYpRilGKyEiIkYpKUkibkdGLyMiIihGI0Y2RjY=
NiQiIiUsJCosIyIiIiIiKkYnKSIiIyNGJyIiJEYnKUkibUc2IkYjRicsJiIjUyEiIiomIiRcJkYnJkkicEdGLzYjRiNGJ0YnRicpSSJuR0YvIyIjNUYsRjJGMg==
NiQiIiYsJCosIyIjSyIiJCIiIilJIm5HNiIjIiM1RighIiIpIiIjI0YpRihGKSlJIm1HRixGI0YpJkkicEdGLDYjIiIlRilGLw==
NiQiIicsJCosIyIjPyIiJCIiIilJIm5HNiIjIiM4RighIiIpIiIjI0YpRihGKSlJIm1HRixGI0YpJkkicEdGLDYjIiIlRilGLw==
NiQiIigsJCosIyIkYyQiI1giIiIpSSJuRzYiIyIjOyIiJCEiIikiIiMjRilGL0YpKUkibUdGLEYjRikmSSJwR0YsNiMiIiVGKUYw
And we get new slopes, yet at the same m powers given in the second sequence;
here we need that the term m^3 is negative as it will dominate in the regime when Ai' is negative,
therefore we have to choose p[4]>2/9 (note that this also makes m^5 and m^7 multiplied by Ai' positive)
ls,li := getslopes(mnmaxB,Mord);
NiQ3JSMhIiYiIz0jISIiIiIjRig3JiIiISIiJCIiJiIiKA==
Draw the new conveux hull in blue
for i from 1 to nops(ls) do
ls[i];li[i];
tt[i] := plot((mnmaxB[li[i]]-ls[i]*li[i])+ls[i]*m,m=li[i]..li[i+1],color=colors[i mod nops(colors)+1],linestyle=styles[i mod nops(styles)+1],legend=[typeset("Slope: ", ls[i])],legendstyle=[location=right]):
end:
Pconvblue := seq(tt[i],i=1..nops(ls)):
#display(%);
Plot the difference to before:
Only the solid circle on the top-left disappeared.
P4bdiffshort := pointplot([0,-4/3],labels=["m deg", "n deg"], symbolsize=15, symbol=solidcircle, color=blue):
display(Pconvblue,P4bdiffshort,P4b2,myview,myoptionsUp,LegendSize);
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Lemma 8
We want to prove
for integers 0 <= j < k <= l <= 2*n,
k-j even,
and n large (here >= 10) that
q(l,j,2*n)/(j+1) >= q(l,k,2*n)/(k+1);
MSomLUkicUc2IjYlSSJsR0YmSSJrR0YmLCQqJiIiIyIiIkkibkdGJkYtRi1GLSwmRilGLUYtRi0hIiIqJi1GJTYlRihJImpHRiZGKkYtLCZGNEYtRi1GLUYw
It suffices to prove that the following is non-negative
LLq := q(l,m-1,2*n)/m-q(l,m+1,2*n)/(m+2);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRMTHFHRigsJiomLUkicUdGKDYlSSJsR0YoLCZJIm1HRigiIiJGNiEiIiwkKiYiIiNGNkkibkdGKEY2RjZGNkY1RjdGNiomLUYxNiVGMywmRjVGNkY2RjZGOEY2LCZGNUY2RjpGNkY3Rjc3I0Yu
This is the recurrence holding for m non-negative
qrec := q(l,m,2*n) = (l-m+1)/(l-m+2)*q(l+1,m-1,2*n)
+ (l-m+2)/(l+m+2)*q(l+1,m+1,2*n)
+ 1/(l-m+4)*q(l+2,m-2,2*n)
+ 1/(l+m+2)*q(l+3,m-1,2*n);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVxcmVjR0YoLy1JInFHRig2JUkibEdGKEkibUdGKCwkKiYiIiMiIiJJIm5HRihGN0Y3LCoqKCwoRjJGN0YzISIiRjdGN0Y3LChGMkY3RjNGPEY2RjdGPC1GMDYlLCZGMkY3RjdGNywmRjNGN0Y3RjxGNEY3RjcqKEY9RjcsKEYyRjdGM0Y3RjZGN0Y8LUYwNiVGQCwmRjNGN0Y3RjdGNEY3RjcqJiwoRjJGN0YzRjwiIiVGN0Y8LUYwNiUsJkYyRjdGNkY3LCZGM0Y3RjZGPEY0RjdGNyomRkNGPC1GMDYlLCZGMkY3IiIkRjdGQUY0RjdGNzcjRi4=
These relations are used in the induction step
qup := subs(m=m+1,isolate(LLq,q(l,m-1,2*n)));
qdown := subs(m=m-1,isolate(LLq,q(l,m+1,2*n)));
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRxdXBHRigvLUkicUdGKDYlSSJsR0YoSSJtR0YoLCQqJiIiIyIiIkkibkdGKEY3RjcqKC1GMDYlRjIsJkYzRjdGNkY3RjRGNywmRjNGNyIiJEY3ISIiLCZGM0Y3RjdGN0Y3NyNGLg==
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZxZG93bkdGKC8tSSJxR0YoNiVJImxHRihJIm1HRigsJComIiIjIiIiSSJuR0YoRjdGNyooLUYwNiVGMiwmRjNGN0Y2ISIiRjRGNywmRjNGN0Y3Rj1GPSwmRjNGN0Y3RjdGNzcjRi4=
Now, we perform an induction on l, with l=2*n being the base case.
We start by applying the recurrence to LLq.
qe1 := collect(subs(subs(m=m-1,qrec),subs(m=m+1,qrec),LLq),q,factor);
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
Now all l's are larger than m, so we can use the inductive hypothesis
Here we use the induction hypothesis and get the rational functions R1, R2, R3.
The idea is to change the parameter m such that all with the same l parameter have the same q.
(Note that if m=2, then the term q(l+2,m-3,2*n) is actually 0; but our qup-shift makes it 0)
qe2 := collect(subs(
subs(l=l+1,m=m-2,qup),
subs(l=l+1,m=m+2,qdown),
subs(l=l+2,m=m-3,qup),
subs(l=l+3,m=m-2,qup),
qe1), q,factor);
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
Next, we check if coefficient of q(l+1,m,2n) is non-negative
the denominator is positive, so we only deal with the numerator
coeff(qe2,q(l+1,m,2*n));
numer(%);
simplify(subs(m=a,l=a+b,%));
LCQqNCIiIyIiIixEKihGJEYlKUkibEc2IiIiJEYlKUkibUdGKkYkRiVGJSooIiIlRiUpRilGJEYlKUYtRitGJSEiIiooRiRGJUYpRiUpRi1GL0YlRiUqKEYvRiVGKEYlRi1GJUYlKihGK0YlRjBGJUYsRiVGJSooIiInRiVGKUYlRjFGJUYyKiZGK0YlRjRGJUYlKiZGK0YlRihGJUYlKigiI0FGJUYwRiVGLUYlRiUqKCIjNkYlRilGJUYsRiVGMiomRi9GJUYxRiVGJSomIiM+RiVGMEYlRiUqKCIjSUYlRilGJUYtRiVGJSomIiM7RiVGLEYlRjIqJiIjUEYlRilGJUYlKiZGL0YlRi1GJUYlIiNARiVGJSwoRilGJUYtRiVGJUYlRjJGLUYyLChGKUYlRi1GMkYlRiVGMiwmRi1GJUYkRiVGMiwoRilGJUYtRjJGK0YlRjIsJkYtRiVGJUYlRjIsKEYpRiVGLUYlRitGJUYyRiU=
LEQqKCIiJSIiIilJImxHNiIiIiRGJSlJIm1HRigiIiNGJUYlKigiIilGJSlGJ0YsRiUpRitGKUYlISIiKihGJEYlRidGJSlGK0YkRiVGJSooRi5GJUYmRiVGK0YlRiUqKCIiJ0YlRi9GJUYqRiVGJSooIiM3RiVGJ0YlRjBGJUYxKiZGNkYlRjNGJUYlKiZGNkYlRiZGJUYlKigiI1dGJUYvRiVGK0YlRiUqKCIjQUYlRidGJUYqRiVGMSomRi5GJUYwRiVGJSomIiNRRiVGL0YlRiUqKCIjZ0YlRidGJUYrRiVGJSomIiNLRiVGKkYlRjEqJiIjdUYlRidGJUYlKiZGLkYlRitGJUYlIiNVRiU=
LDIqJiIiKSIiIilJImFHNiIiIiVGJUYlKihGKUYlKSwmSSJiR0YoRiUiIiRGJSIiI0YlKUYnRi5GJUYlKiYsKiomRilGJSlGLUYuRiVGJSomIiNJRiUpRi1GL0YlRiUqJiIjJSlGJUYtRiVGJSIjbUYlRiUpRidGL0YlRiUqJiwqKiZGJEYlRjRGJUYlKiYiI2lGJUY3RiVGJSomIiRPIkYlRi1GJUYlIiMjKUYlRiVGJ0YlRiUqJiIiJ0YlRjRGJUYlKiYiI1FGJUY3RiVGJSomIiN1RiVGLUYlRiUiI1VGJQ==
Then, we use the following simple bounds, arising by taking just the first term on the right-hand side of qrec;
Note that we DO NOT have equality here but a >= sign;
however, for the subsitution process we use it like that.
qrecUpBound := q(l,m,2*n) = coeff(rhs(qrec),q(l+1,m-1,2*n))*q(l+1,m-1,2*n);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSxxcmVjVXBCb3VuZEdGKC8tSSJxR0YoNiVJImxHRihJIm1HRigsJComIiIjIiIiSSJuR0YoRjdGNyooLChGMkY3RjMhIiJGN0Y3RjcsKEYyRjdGM0Y7RjZGN0Y7LUYwNiUsJkYyRjdGN0Y3LCZGM0Y3RjdGO0Y0Rjc3I0Yu
We will use this bound
subs(l=l+1,qrecUpBound);
Ly1JInFHNiI2JSwmSSJsR0YlIiIiRilGKUkibUdGJSwkKiYiIiNGKUkibkdGJUYpRikqKCwoRihGKUYqISIiRi1GKUYpLChGKEYpRipGMSIiJEYpRjEtRiQ2JSwmRihGKUYtRiksJkYqRilGKUYxRitGKQ==
Now the next 2 are the same
qe3 := collect(subs(
subs(l=l+1,qrecUpBound),
qe2), q,factor);
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
Then, we check if coefficient of q(l+2,m-1,2n) is non-negative
the denominator is positive, so we only deal with the numerator
coeff(qe3,q(l+2,m-1,2*n));
numer(%);
simplify(subs(m=a+2,l=a+b+2,%));
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
Finally, in the same way as before, we use the following bound, arising by taking just the second term on the right-hand side of qrec;
Note that we DO NOT have equality here but a >= sign;
however, for the subsitution process we use it like that.
qrecUpBound2 := q(l,m,2*n) = coeff(rhs(qrec),q(l+1,m+1,2*n))*q(l+1,m+1,2*n);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS1xcmVjVXBCb3VuZDJHRigvLUkicUdGKDYlSSJsR0YoSSJtR0YoLCQqJiIiIyIiIkkibkdGKEY3RjcqKCwoRjJGN0YzISIiRjZGN0Y3LChGMkY3RjNGN0Y2RjdGOy1GMDYlLCZGMkY3RjdGNywmRjNGN0Y3RjdGNEY3NyNGLg==
We will use this bound
subs(l=l+2,m=m-1,qrecUpBound2);
Ly1JInFHNiI2JSwmSSJsR0YlIiIiIiIjRiksJkkibUdGJUYpRikhIiIsJComRipGKUkibkdGJUYpRikqKCwoRihGKUYsRi0iIiZGKUYpLChGKEYpRixGKSIiJEYpRi0tRiQ2JSwmRihGKUY1RilGLEYuRik=
Now all are the same
qe4 := collect(subs(
subs(l=l+2,m=m-1,qrecUpBound2),
qe3), q,factor);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRxZTRHRigsJCo2IiIjIiIiLGZvKihGMEYxKUkibEdGKCIiJkYxKUkibUdGKCIiJEYxRjEqKCIiKUYxKUY1IiIlRjEpRjhGPUYxISIiKigiIzdGMSlGNUY5RjEpRjhGNkYxRjEqKEY7RjEpRjVGMEYxKUY4IiInRjFGPyooRjBGMUY1RjEpRjgiIihGMUYxKihGPUYxRjRGMSlGOEYwRjFGMSooIiIqRjFGPEYxRjdGMUYxKigiI19GMUZCRjFGPkYxRj8qKCIjaUYxRkVGMUZDRjFGMSooIiNDRjFGNUYxRkZGMUY/KiRGSUYxRjEqKCIjWkYxRjxGMUZMRjFGMSooIiNMRjFGQkYxRjdGMUY/KigiI3ZGMUZFRjFGPkYxRj8qKCIjdEYxRjVGMUZDRjFGMSomRkFGMUZGRjFGPyomRjBGMUY0RjFGPyooRkpGMUY8RjFGOEYxRjEqKCIkKj5GMUZCRjFGTEYxRjEqKCIkQSNGMUZFRjFGN0YxRj8qKCIjWEYxRjVGMUY+RjFGMSokRkNGMUYxKiYiI0FGMUY8RjFGPyooIiNlRjFGQkYxRjhGMUYxKigiJG4kRjFGRUYxRkxGMUYxKigiJEwkRjFGNUYxRjdGMUY/KiYiIyMqRjFGPkYxRjEqJkZbcEYxRkJGMUY/KigiJHMiRjFGRUYxRjhGMUYxKigiJCJHRjFGNUYxRkxGMUYxKiYiJEYiRjFGN0YxRj8qJiIkIT1GMUZFRjFGPyooIiQ5I0YxRjVGMUY4RjFGMSomIiNxRjFGTEYxRjEqJiIkaSJGMUY1RjFGPyomIiMkKkYxRjhGMUYxIiNhRj9GMSwoRjVGMUY4RjFGMUYxRj9GTEY/LChGNUYxRjhGP0YxRjFGPywmRjhGMUYwRjFGPyksKEY1RjFGOEY/RjlGMUYwRj8sJkY4RjFGMUYxRj8pLChGNUYxRjhGMUY5RjFGMEY/LUkicUdGKDYlLCZGNUYxRjlGMUY4LCQqJkYwRjFJIm5HRihGMUYxRjFGMTcjRi4=
So we want to show that the factor of q(l+3,m,2*n) is positive;
note that l >= m >= 1:
therefore we set m=a+2 and l=a+b+2 in order to have a,b>=0 independently;
and we see that all terms have non-negative coefficients, which proves the claim.
qe5 := simplify(subs(q(l+3,m,2*n)=1,m=a+1,l=a+b+1,qe4));
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
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
JSFH
JSFH