M = 13, L = 4, via "hybrid resultant method"

> restart;

> st:=time():

Initializations and routines for various steps in the computation

> read "/home/fac/little/PuertoRico/Filters/CompFileXX.map"; 

Warning, new definition for norm

Warning, new definition for trace

Warning, `n` in call to `seq` is not local

Warning, `n` in call to `seq` is not local

Warning, `n` in call to `seq` is not local

Warning, `n` in call to `seq` is not local

> QM13L4:=quads(4,13):

> GBLexM13L4:=gbasis(subs(m[0]=1,QM13L4),plex(seq(m[27-j],j=1..26))):

> L:=4: M:=13: K:=1:

> read "/home/fac/little/PuertoRico/Filters/CompScratchFile.map":

> linearEqs:=subs(m[0]=1,[seq(m[j]-mm[j],j=19..26),meq]):

The parametrization of the rational scroll

> sublist1:={m[1]=t,m[2]=t^2,m[3]=t^3,m[4]=t^4,m[5]=t^5,m[6]=t^6,m[7]=t^7,m[8]=t^8,m[9]=t^9,m[10]=t^10,m[11]=a,m[13]=b,m[15]=c,m[17]=d,m[19]=e,m[21]=f,m[23]=g,m[25]=h}:

> SGB:=subs(sublist1,GBLexM13L4):

> sublist2:={m[12]=solve(SGB[10],m[12]),m[14]=solve(SGB[11],m[14]),m[16]=solve(SGB[12],m[16]),m[18]=solve(SGB[13],m[18]),m[20]=solve(SGB[14],m[20]),m[22]=solve(SGB[15],m[22]),m[24]=solve(SGB[16],m[24]),m[26]=solve(SGB[17],m[26])};

[Maple Math]
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[Maple Math]

> RES:=collect(subs(sublist1 union sublist2,linearEqs),[a,b,c,d,e,f,g,h]):

RES[1] has a,b,c,d,e

RES[2] has a,b,c,d,e

RES[3] has a,b,c,d,e,f

RES[4] has a,b,c,d,e,f

RES[5] has a,b,c,d,e,f,g

RES[6] has a,b,c,d,e,f,g

RES[7] has a,b,c,d,e,f,g,h

RES[8] has a,b,c,d,e,f,g,h

RES[9] has a,b,c,d

> Rh:=resultant(RES[7],RES[8],h):

> Rg1:=resultant(RES[5],Rh,g):

> Rg2:=resultant(RES[6],Rh,g):

At this point, have 7 polys with a,b,c,d,e,f

> Rf1:=resultant(RES[4],Rg1,f):

> Rf2:=resultant(RES[3],Rg2,f):

> Rf3:=resultant(RES[3],RES[4],f):

At this point, have > 6 polys with a,b,c,d,e

> Re1:=resultant(RES[1],Rf1,e):

> Re2:=resultant(RES[2],Rf2,e):

> Re3:=resultant(RES[1],RES[2],e):

> Re4:=resultant(Rf3,Rf2,e):

At this point, have > 5 polys with a,b,c,d

> Rd1:=resultant(Re1,RES[9],d):

> Rd2:=resultant(Re3,RES[9],d):

> Rd3:=resultant(Re3,Re4,d):

> Rd4:=resultant(Re2,RES[9],d):

At this point, have 4 polys with a,b,c

> Rc1:=resultant(Rd1,Rd2,c):

> Rc2:=resultant(Rd1,Rd3,c):

> Rc3:=resultant(Rd2,Rd4,c):

Eliminate b, then a

> Rb1:=resultant(Rc1,Rc2,b):

> Rb2:=resultant(Rc1,Rc3,b):

> RFinal1:=factor(resultant(Rb1,Rb2,a)):

Apart from constants and even powers of t -9, expect only one factor here to be

symmetric under t -> 18 - t -- that's the one we're looking for:

> for i to nops(RFinal1) do if simplify(subs(t=18-t,op(i,RFinal1))) = op(i,RFinal1) then lprint(i) fi od: 

1

3

8

> symbolicpart:=time()-st;

[Maple Math]

> st:=time():

> realone:=op(8,RFinal1):

> Digits:=130:

> rts:=[fsolve(realone,t,complex)]:

> for i to nops(rts) do if Im(rts[i]) = 0 then lprint(evalf(rts[i],20)) fi od: 

.20895158162668901418e-3

1.0030837240903281225

2.0203160065661641464

3.0809913794372601186

4.2456525180356737537

6.5605001085339155072

6.8508303001264915098

7.6047190731545017963

10.395280926845498204

11.149169699873508490

11.439499891466084493

13.754347481964326246

14.919008620562739881

15.979683993433835854

16.996916275909671878

17.999791048418373311

> numericpart:=time()-st;

[Maple Math]