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Question: Can I trust the diffrent engines of PolynomialSystem

Question:Can I trust the diffrent engines of PolynomialSystem

Saalehorizontale 35 Maple 2024
polynomial solvetools
December 18 2024
0

Hey Guys, 

I have to solve multiple system of equations under some restrictions given as inequalities. Sometimes solve is not able to find the result in houres so I tryd to break the problem in half. So in the first step I just want to solve my 8 polynomial equations with 8 variables and in a second step I want so take the solutions, bring them together with the set of inequalities and solve it again. Since also some sets of equations are to hard for the simple solve command I got the advice from people of this plattform to try PolynomialSystem with the diffrent engines. However I have the feeling they make misstakes and now Im not sure If I can trust my results. 

Attached you can find a file with an example. In the beginning I solve equations and restrictions together and there is a solutions. Then I tryd to solve only the equations with PolynomialSystem and the the four known engines and the eniges traditional and backsolve dont find the solution which as we saw before exist. When a soultions holds under restrictions it should always appear if I omit the restirctions. When I use the enige triade and groebner then the right solution is there. 
However in some other cases it feels the other way round.
So to me it looks like no matter which engine I take, I can never 100% trust my results. Did I something wrong? Whats the reason for those mistakes? Furthermore backsolve gives me 7 solutions, but solutions 2 and 7 are the same. I also recognized, that there is a diffrence between putting in the variable vars as a list or a set. What happens, if I dont specify which engine should be used?

I am happy about any advice. Thank you in advance.

Regards

Felix

> 

restart; equations := {-y*(m-p) = 0, ((-x-y+1)*k+x)*n+s*y-t = 0, (k-x-y)*t-k*p+y = 0, (-m+n+y)*x+m-1 = 0, -(x+y-1)*(p-t)*k+(-x-y+1)*t+x*p = 0, y^2+(-m-1)*y+1+x*(p-1) = 0, (-x-y+1)*t+(-m+1)*x+y*n+m-1 = 0, -k*n+s*x = 0}; restrictions := {0 < k, 0 < m, 0 < s, 0 < x, 0 < y, 0 < n+(t-1)*p, 0 < (m*y-1)*n+(1-p)*(m*x-m+1), 0 < (m*x-m-t+1)*p+m*y*(t-n), 1 < x+y, k < 1, m < 1, s < t, t < 1}; vars := indets(equations); evalf(solve(`union`(equations, restrictions), vars)); Sol_w := SolveTools:-PolynomialSystem(equations, vars); Sol_traditional := SolveTools:-PolynomialSystem(equations, vars, engine = traditional); nops([Sol_traditional]); Sol_backsolve := SolveTools:-PolynomialSystem(equations, vars, engine = backsolve); nops([Sol_backsolve]); Sol_triade_1 := SolveTools:-PolynomialSystem(equations, vars, engine = triade); nops([Sol_triade_1]); Sol_groebner := SolveTools:-PolynomialSystem(equations, vars, engine = groebner); nops([Sol_groebner])

{-y*(m-p) = 0, ((-x-y+1)*k+x)*n+s*y-t = 0, (k-x-y)*t-k*p+y = 0, (-m+n+y)*x+m-1 = 0, -(x+y-1)*(p-t)*k+(-x-y+1)*t+x*p = 0, y^2+(-m-1)*y+1+x*(p-1) = 0, (-x-y+1)*t+(-m+1)*x+y*n+m-1 = 0, -k*n+s*x = 0}

  

{k, m, n, p, s, t, x, y}

  

{k = 0.536796024e-1, m = .241141717, n = .54019322, p = .241141717, s = 0.35770767e-1, t = .4477103163, x = .8106439941, y = .6370663217}

  

{k = 1, m = m, n = 1, p = 0, s = 1, t = 1, x = 1, y = 0}, {k = 1, m = 1, n = 0, p = 1, s = t, t = t, x = 0, y = 1}, {k = k, m = 1, n = 0, p = 1, s = 1, t = 1, x = 0, y = 1}, {k = 1, m = 1, n = 0, p = 1, s = 0, t = 0, x = x, y = 1}, {k = 1/3, m = -1, n = 3, p = -1, s = 2, t = 2, x = 1/2, y = 0}, {k = 1/3, m = -1, n = 2, p = -1, s = 2/3, t = 2, x = 1, y = -1}, {k = -(1/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4+(2/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3+(16/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+(1/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-7/9, m = (4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(20/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(17/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+21*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-28/3, n = (11/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(53/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(55/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+(152/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-61/3, p = (4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(20/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(17/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+21*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-28/3, s = -(8/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4+(40/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3+(35/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2-(127/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)+58/9, t = (1/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(8/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+(7/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)+1/3, x = (1/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(8/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+(4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)+4/3, y = RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)}

  

{k = 1, m = m, n = 1, p = 0, s = 1, t = 1, x = 1, y = 0}, {k = 1/3, m = -1, n = 3, p = -1, s = 2, t = 2, x = 1/2, y = 0}, {k = k, m = 1, n = 0, p = 1, s = 1, t = 1, x = 0, y = 1}, {k = 1, m = 1, n = 0, p = 1, s = s, t = s, x = 0, y = 1}, {k = 1/3, m = -1, n = 2, p = -1, s = 2/3, t = 2, x = 1, y = -1}, {k = 1, m = 1, n = 0, p = 1, s = 0, t = 0, x = x, y = 1}

  

6

  

{k = 1, m = m, n = 1, p = 0, s = 1, t = 1, x = 1, y = 0}, {k = 1/3, m = -1, n = 3, p = -1, s = 2, t = 2, x = 1/2, y = 0}, {k = k, m = 1, n = 0, p = 1, s = 1, t = 1, x = 0, y = 1}, {k = 1, m = 1, n = 0, p = 1, s = s, t = s, x = 0, y = 1}, {k = 1/3, m = -1, n = 2, p = -1, s = 2/3, t = 2, x = 1, y = -1}, {k = 1, m = 1, n = 0, p = 1, s = 0, t = 0, x = x, y = 1}, {k = 1/3, m = -1, n = 3, p = -1, s = 2, t = 2, x = 1/2, y = 0}

  

7

  

{k = 1, m = 1, n = 0, p = 1, s = 0, t = 0, x = x, y = 1}, {k = 1, m = m, n = 1, p = 0, s = 1, t = 1, x = 1, y = 0}, {k = k, m = 1, n = 0, p = 1, s = 1, t = 1, x = 0, y = 1}, {k = 1, m = 1, n = 0, p = 1, s = t, t = t, x = 0, y = 1}, {k = -(1/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4+(2/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3+(16/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+(1/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-7/9, m = (4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(20/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(17/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+21*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-28/3, n = (11/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(53/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(55/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+(152/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-61/3, p = (4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(20/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(17/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+21*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-28/3, s = -(8/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4+(40/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3+(35/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2-(127/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)+58/9, t = (1/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(8/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+(7/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)+1/3, x = (1/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(8/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+(4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)+4/3, y = RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)}, {k = 1/3, m = -1, n = 2, p = -1, s = 2/3, t = 2, x = 1, y = -1}, {k = 1/3, m = -1, n = 3, p = -1, s = 2, t = 2, x = 1/2, y = 0}

  

7

  

{k = 1, m = m, n = 1, p = 0, s = 1, t = 1, x = 1, y = 0}, {k = 1, m = 1, n = 0, p = 1, s = t, t = t, x = 0, y = 1}, {k = k, m = 1, n = 0, p = 1, s = 1, t = 1, x = 0, y = 1}, {k = 1, m = 1, n = 0, p = 1, s = 0, t = 0, x = x, y = 1}, {k = 1/3, m = -1, n = 3, p = -1, s = 2, t = 2, x = 1/2, y = 0}, {k = 1/3, m = -1, n = 2, p = -1, s = 2/3, t = 2, x = 1, y = -1}, {k = -(1/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4+(2/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3+(16/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+(1/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-7/9, m = (4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(20/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(17/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+21*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-28/3, n = (11/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(53/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(55/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+(152/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-61/3, p = (4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(20/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(17/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+21*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-28/3, s = -(8/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4+(40/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3+(35/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2-(127/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)+58/9, t = (1/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(8/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+(7/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)+1/3, x = (1/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-(4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3-(8/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2+(4/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)+4/3, y = RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)}

  

7

 (1)
> 

evalf(allvalues({k = -(1/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4+2*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3*(1/9)+16*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2*(1/9)+(1/9)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-7/9, m = 4*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4*(1/3)-20*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3*(1/3)-17*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2*(1/3)+21*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-28/3, n = 11*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4*(1/3)-53*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3*(1/3)-55*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2*(1/3)+152*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)*(1/3)-61/3, p = 4*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4*(1/3)-20*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3*(1/3)-17*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2*(1/3)+21*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)-28/3, s = -8*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4*(1/9)+40*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3*(1/9)+35*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2*(1/9)-127*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)*(1/9)+58/9, t = (1/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-4*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3*(1/3)-8*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2*(1/3)+7*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)*(1/3)+1/3, x = (1/3)*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^4-4*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^3*(1/3)-8*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)^2*(1/3)+4*RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)*(1/3)+4/3, y = RootOf(_Z^5-4*_Z^4-9*_Z^3+10*_Z^2+6*_Z-5)}))

{k = 0.536796024e-1, m = .241141717, n = .54019322, p = .241141717, s = 0.35770767e-1, t = .4477103163, x = .8106439941, y = .6370663217}, {k = .7943583912, m = 1.011543377, n = .16794280, p = 1.011543377, s = -.463558437, t = -.4040771797, x = -.287788440, y = .8837112597}, {k = -5.038767243, m = 3.694058367, n = 0.9373027e-1, p = 3.694058367, s = .299187114, t = 2.728412223, x = -1.578565716, y = 5.306977937}, {k = .2033642547, m = -26.40026363, n = -63.64948932, p = -26.40026363, s = 17.99511944, t = -2.562622110, x = -.719307867, y = -.8433142428}, {k = 2.542920564, m = -.546480183, n = 1.84762297, p = -.546480183, s = 1.244592174, t = .7905767063, x = 3.775017982, y = -1.984441276}

 (2)
 

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