A bifurcation problem

Equations:

5z_{1}^{9} − 6z_{1}^{5}z_{2} + z_{1}z_{2}^{4} + 2z_{1}z_{3}; −2z_{1}^{6}z_{2} + 2z_{1}^{2}z_{2}^{3} + 2z_{2}z_{3}; z_{1}^{2} + z_{2}^{2} − 0.265625;

This polynomial system arises from a test for Numerical Bifurcation and it has been extracted from [1]. This exemple is also in the PoSSo test suite.

Characteristics:

• There are 8 real solutions, which are in the box [−0.6, 6]^{2} × [−5, 5]:

\begin{matrix}z_{1} & z_{2} & z_{3} \\ -.5153882032 & .1185133007 & -.01244559884 \\ .5015771103 & .1185133007 & .01238951314 \\ .2619366407 & .4438628124 & -.01319432095 \\ -.2619366407 & .4438628124 & -.01319432095 \\ -.5015771103 & .0 & .01238951314 \\ .5153882032 & .0 & -.01244559884 \\ .0 & -.5153882032 & .0 \\ .0 & .5153882032 & .0 \end{matrix}

• It can be splitted into the subsystems:

{64 z_{1}^{2} − 17, z_{2}, 33554432 z_{3} + 417605}

\{ \begin{matrix}83886080 z_{2}^{8} − 122683392 z_{2}^{6} − 100663296 z_{2}^{5} + 112582656 z_{2}^{4} \\ +53477376 z_{2}^{3} − 22304000 z_{2}^{2} − 7102464 z_{2} + 1046469, \\ 64 z_{1}^{2} + 64 z_{2}^{2} − 17, \\ 262144 z_{3} + 262144 z_{2}^{6} − 471040 z_{2}^{4} + 125120 z_{2}^{2} − 4913 \end{matrix} \}

{64 z_{2}^{2} − 17, z_{3}, z_{1}}

• The number of complex roots is therefore 20.

Example 1:

var := [z[1],z[2],z[3]];
[5*z[1]^9-6*z[1]^5*z[2]+z[1]*z[2]^4+2*z[1]*z[3],
-2*z[1]^6*z[2]+2*z[1]^2*z[2]^3+2*z[2]*z[3], z[1]^2+z[2]^2-0.265625];

• Solution by \underline{J.P. Merlet}

Methode: bisection avec analyse par intervalle utilisant jacobienne et hessienne des equations.

Temps de calcul:1.46s sur SUN ULTRA 1

• Solution by \underline{B. Mourrain}

In maple, suing the function grobner[gsolve], one obtain the decomposition of the system, and solve it by univariate polynomial solving and substitution.

With the packadge \underline{multires} hiding one variable, we obtain the different coordinates of the roots.

See \underline{the maple session}.

References

[1] R.B. Kearfott. Some tests of generalized bisection. ACM Transactions on Mathematical Software, 13(3):197–220, 1987.