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 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 ``` ``````### Trop compliqué, le calcul ne termine pas... :-( e = SymmetricFunctionAlgebra(QQ, basis='elementary') R. = PolynomialRing(QQ,12,order='lex(5),lex(1),lex(1),deglex(5)') asym1 = -e([1]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym2 = +e([2]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym3 = -e([3]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym4 = +e([4]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym5 = -e([5]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) Isym = R.ideal([a1-asym1, a2-asym2, a3-asym3, a4-asym4, a5-asym5]) pexpr = z1^2*(z2*z5+z3*z4)+z2^2*(z1*z3+z4*z5)+z3^2*(z1*z5+z2*z4)+z4^2*(z1*z2+z3*z5)+z5^2*(z1*z4+z2*z3) Ip = R.ideal([p-pexpr]) dexpr = (z1-z2)*(z1-z3)*(z1-z4)*(z2-z3)*(z2-z4)*(z3-z4) Id = R.ideal([d-dexpr]) qexpr = z1*z2 + z2*z3 + z3*z4 + z4*z5 + z1*z5 B = (Isym+Ip+Id).groebner_basis() qexpr.reduce(B) ### Autre tentative... toujours trop compliquée. e = SymmetricFunctionAlgebra(QQ, basis='elementary') R. = PolynomialRing(QQ,11,order='lex(5),lex(1),deglex(5)') asym1 = -e([1]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym2 = +e([2]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym3 = -e([3]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym4 = +e([4]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym5 = -e([5]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) Isym = R.ideal([a1-asym1, a2-asym2, a3-asym3, a4-asym4, a5-asym5]) pexpr = z1^2*(z2*z5+z3*z4)+z2^2*(z1*z3+z4*z5)+z3^2*(z1*z5+z2*z4)+z4^2*(z1*z2+z3*z5)+z5^2*(z1*z4+z2*z3) Ip = R.ideal([p-pexpr]) qexpr = z1*z2 + z2*z3 + z3*z4 + z4*z5 + z1*z5 qexprConj = qexpr.subs({z2:z5,z3:z4,z4:z3,z5:z2}) coef0 = qexpr*qexprConj coef1 = -(qexpr+qexprConj) B = (Isym+Ip).groebner_basis() coef1.reduce(B) coef0.reduce(B) ### Encore une tentative... pas mieux... e = SymmetricFunctionAlgebra(QQ, basis='elementary') R. = PolynomialRing(QQ,11,order='lex(5),lex(1),deglex(5)') asym1 = -e([1]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym2 = +e([2]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym3 = -e([3]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym4 = +e([4]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym5 = -e([5]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) Isym = R.ideal([a1-asym1, a2-asym2, a3-asym3, a4-asym4, a5-asym5]) pexpr = z1^2*(z2*z5+z3*z4)+z2^2*(z1*z3+z4*z5)+z3^2*(z1*z5+z2*z4)+z4^2*(z1*z2+z3*z5)+z5^2*(z1*z4+z2*z3) qexpr = z1*z2 + z2*z3 + z3*z4 + z4*z5 + z1*z5 Iq = R.ideal([q-qexpr]) B = (Isym+Iq).groebner_basis() pexpr.reduce(B) ### Essayons avec de l'algèbre linéaire e = SymmetricFunctionAlgebra(QQ, basis='elementary') R. = PolynomialRing(QQ,10,order='lex(5),deglex(5)') asym1 = -e([1]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym2 = +e([2]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym3 = -e([3]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym4 = +e([4]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) asym5 = -e([5]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5) Isym = R.ideal([a1-asym1, a2-asym2, a3-asym3, a4-asym4, a5-asym5]) B = Isym.groebner_basis() fexpr = z1^2*z2 + z2^2*z3 + z3^2*z4 + z4^2*z5 + z1*z5^2 pexpr = z1^2*(z2*z5 + z3*z4) + z2^2*(z1*z3 + z4*z5) + z3^2*(z1*z5 + z2*z4) + z4^2*(z1*z2 + z3*z5) + z5^2*(z1*z4 + z2*z3) qexpr = z1*z2 + z2*z3 + z3*z4 + z4*z5 + z1*z5 dexpr = (z1-z2)*(z1-z3)*(z1-z4)*(z1-z5)*(z2-z3)*(z2-z4)*(z2-z5)*(z3-z4)*(z3-z5)*(z4-z5) def debase(poly): r = poly.reduce(B) return [r.coefficient({z1:0,z2:i2,z3:i3,z4:i4,z5:i5}) for i2 in range(2) for i3 in range(3) for i4 in range(4) for i5 in range(5)] qtab = [debase(qexpr^i) for i in range(12)] pdtab = [debase(pexpr^i*dexpr^j) for i in range(6) for j in range(2)] qmat = matrix(R, 12, 120, qtab) pdmat = matrix(R, 12, 120, pdtab) cols = [0, 10, 105, 84, 109, 119, 57, 91, 111, 90, 110, 15] qsubmat = qmat[range(12),cols] pdsubmat = pdmat[range(12),cols] qsubdet = qsubmat.determinant() complements = [[i for i in range(12) if i!=j] for j in range(12)] results = [[None for j in range(12)] for i in range(12)] for i in range(12): for j in range(12): print (i,j) results[i][j] = (-1)^(i+j)*(qsubmat[complements[j],complements[i]].determinant()) qsubadj = matrix(R, 12, 12, results) tmp = [sum([pdsubmat[1][k]*qsubadj[k][j] for k in range(12)]) for j in range(12)] subfactor = -tmp[11]/1280 qsubdetfact = R(qsubdet/subfactor) donq = [R(pol/subfactor)/qsubdetfact for pol in tmp] ``````