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I am trying to find a Poisson bracket on an algebra, and need to find a solution to a system of equations. The system of equations is very complicated, with more than 10000 equations and 60 variables.

The following is a part of the system of the equations: \begin{align} & 6 z_{2} z_{10} - 12 z_{4} + 6 z_{3} z_{13} + 12 z_{4} z_{23} - 12 z_{4} z_{44} + 6 z_{7} z_{41} - 12 z_{4} z_{52} + 12 z_{12} z_{48} - 12 z_{8} z_{54}= 0, \\ &3 z_{2} - 2 z_{7} - 4 z_{4} z_{6} - 4 z_{2} z_{23} - 4 z_{4} z_{22} + 4 z_{12} z_{15} + 2 z_{7} z_{23} + 4 z_{2} z_{29} - 4 z_{8} z_{24} + 4 z_{3} z_{30} + 4 z_{2} z_{44} - 4 z_{7} z_{52} + 4 z_{10} z_{50} + 4 z_{16} z_{48} - 4 z_{11} z_{54} + 4 z_{13} z_{55} + 4 z_{28} z_{41} - 4 z_{41} z_{50}= 0, \\ &2 z_{10} z_{23} - z_{41} - 4 z_{4} z_{9} - 2 z_{10} + 4 z_{2} z_{38} - 4 z_{4} z_{36} + 4 z_{3} z_{39} - 4 z_{8} z_{37} + 4 z_{12} z_{33} + 4 z_{2} z_{47} + 4 z_{7} z_{47} - 4 z_{10} z_{44} + 4 z_{18} z_{48} - 4 z_{14} z_{54} + 4 z_{13} z_{56} + 4 z_{29} z_{41} - 4 z_{41} z_{52}= 0, \\ &2 z_{13} z_{23} - 4 z_{4} z_{15} - 3 z_{13} + 4 z_{2} z_{39} - 4 z_{4} z_{37} + 4 z_{2} z_{48} + 4 z_{3} z_{49} + 4 z_{12} z_{40} + 4 z_{7} z_{48} - 4 z_{13} z_{44} - 4 z_{8} z_{51} + 4 z_{10} z_{54} - 4 z_{13} z_{52} + 4 z_{19} z_{48} + 4 z_{13} z_{57} - 4 z_{17} z_{54} + 4 z_{30} z_{41} - 4 z_{41} z_{54}= 0, \\ &2 z_{50} + 4 z_{2} z_{6} - 2 z_{6} z_{7} - 4 z_{7} z_{22} + 2 z_{10} z_{21} + 4 z_{15} z_{16} - 4 z_{11} z_{24} + 2 z_{13} z_{25} - 4 z_{21} z_{41} + 4 z_{28} z_{44} - 4 z_{23} z_{50} + 8 z_{29} z_{50} - 4 z_{28} z_{52} + 8 z_{30} z_{55} - 4 z_{32} z_{54} + 4 z_{48} z_{59}= 0, \\ &4 z_{2} z_{9} - 2 z_{6} z_{10} + 2 z_{15} z_{18} - 2 z_{14} z_{24} + 2 z_{13} z_{26} - 2 z_{7} z_{36} + 4 z z_{48} - 2 z_{11} z_{37} + 2 z_{16} z_{33} - 4 z_{22} z_{41} + 4 z_{28} z_{47} + 4 z_{30} z_{56} - 4 z_{34} z_{54} + 4 z_{38} z_{50} + 4 z_{39} z_{55}= 0, \\ &4 z_{2} z_{15} - z_{54} - z_{30} - 2 z_{6} z_{13} + 2 z_{10} z_{24} + 2 z_{15} z_{19} - 2 z_{13} z_{22} + 2 z_{13} z_{27} - 2 z_{17} z_{24} - 2 z_{7} z_{37} + 4 z_{1} z_{48} + 2 z_{16} z_{40} - 2 z_{11} z_{51} - 4 z_{24} z_{41} + 4 z_{28} z_{48} - 4 z_{30} z_{52} + 4 z_{29} z_{54} + 4 z_{30} z_{57} - 4 z_{35} z_{54} + 4 z_{39} z_{50} + 4 z_{49} z_{55}= 0, \\ &2 z_{47} - 4 z_{9} z_{10} + 4 z_{2} z_{31} + 2 z_{7} z_{31} - 2 z_{10} z_{36} - 4 z_{14} z_{37} + 4 z_{18} z_{33} + 2 z_{13} z_{43} + 4 z_{20} z_{48} - 4 z_{23} z_{47} + 8 z_{29} z_{47} - 4 z_{36} z_{41} - 4 z_{38} z_{44} + 4 z_{38} z_{52} + 8 z_{39} z_{56} - 4 z_{42} z_{54}= 0, \\ &z_{48} - z_{39} - 2 z_{9} z_{13} - 2 z_{10} z_{15} + 4 z_{2} z_{33} + 2 z_{7} z_{33} - 2 z_{13} z_{36} + 2 z_{19} z_{33} - 2 z_{17} z_{37} + 2 z_{18} z_{40} + 2 z_{13} z_{46} - 2 z_{14} z_{51} + 4 z_{29} z_{48} + 4 z_{30} z_{47} - 4 z_{37} z_{41} - 4 z_{39} z_{44} + 4 z_{38} z_{54} + 4 z_{39} z_{57} - 4 z_{45} z_{54} + 4 z_{49} z_{56}= 0, \\ &4 z_{2} z_{40} - 4 z_{13} z_{15} - 2 z_{49} + 2 z_{7} z_{40} - 4 z_{13} z_{37} + 4 z_{5} z_{48} + 4 z_{19} z_{40} + 2 z_{10} z_{51} - 4 z_{17} z_{51} + 2 z_{13} z_{58} + 8 z_{30} z_{48} - 4 z_{41} z_{51} + 8 z_{39} z_{54} - 4 z_{44} z_{49} - 4 z_{49} z_{52} + 8 z_{49} z_{57} - 4 z_{53} z_{54}= 0, \\ &2 z_{4} - z_{2} z_{10} - z_{3} z_{13} - 2 z_{4} z_{23} + 2 z_{4} z_{44} - z_{7} z_{41} + 2 z_{4} z_{52} - 2 z_{12} z_{48} + 2 z_{8} z_{54}= 0, \\ &\frac{3 z_{8}}{2} - z_{2} z_{14} - z_{3} z_{17} - 2 z_{8} z_{23} + 2 z_{8} z_{44} - z_{11} z_{41} + 2 z_{12} z_{47} + 2 z_{4} z_{56} + 2 z_{8} z_{57}= 0, \\ & z_{2} z_{10} - 2 z_{4} + z_{3} z_{13} + 2 z_{4} z_{23} - 2 z_{4} z_{44} + z_{7} z_{41} - 2 z_{4} z_{52} + 2 z_{12} z_{48} - 2 z_{8} z_{54}= 0, \\ & z_{2} z_{14} - \frac{3 z_{8}}{2} + z_{3} z_{17} + 2 z_{8} z_{23} - 2 z_{8} z_{44} + z_{11} z_{41} - 2 z_{12} z_{47} - 2 z_{4} z_{56} - 2 z_{8} z_{57}= 0, \\ &\frac{3 z_{12}}{2} - z_{2} z_{18} - z_{3} z_{19} - 2 z_{12} z_{23} - z_{16} z_{41} + 2 z_{8} z_{50} - 2 z_{4} z_{55} + 2 z_{12} z_{52} + 2 z_{12} z_{57}= 0, \\ & z_{2} z_{18} - \frac{3 z_{12}}{2} + z_{3} z_{19} + 2 z_{12} z_{23} + z_{16} z_{41} - 2 z_{8} z_{50} + 2 z_{4} z_{55} - 2 z_{12} z_{52} - 2 z_{12} z_{57}= 0, \\ &12 z_{21} z_{29} - 3 z_{21} - 12 z_{22} z_{28} + 12 z_{25} z_{30} + 12 z_{6} z_{50} - 12 z_{24} z_{32} - 12 z_{21} z_{44} + 12 z_{15} z_{59}= 0, \\ & z_{6} - 4 z_{22} + 8 z z_{15} - 4 z_{6} z_{29} + 4 z_{9} z_{28} + 4 z_{22} z_{23} + 8 z_{26} z_{30} + 4 z_{6} z_{52} - 8 z_{24} z_{34} + 8 z_{9} z_{50} + 4 z_{21} z_{38} + 4 z_{25} z_{39} - 4 z_{28} z_{36} - 8 z_{22} z_{44} - 4 z_{21} z_{47} - 4 z_{32} z_{37} + 4 z_{33} z_{59}= 0, \\ &8 z_{1} z_{15} - 6 z_{24} - 4 z_{6} z_{30} + 4 z_{15} z_{28} + 4 z_{23} z_{24} - 8 z_{22} z_{30} + 8 z_{24} z_{29} + 8 z_{27} z_{30} - 8 z_{24} z_{35} + 4 z_{6} z_{54} + 4 z_{21} z_{39} + 8 z_{15} z_{50} - 4 z_{28} z_{37} - 8 z_{24} z_{44} - 4 z_{21} z_{48} + 4 z_{25} z_{49} - 4 z_{32} z_{51} + 4 z_{40} z_{59}= 0, \\ &4 z_{9} - 3 z_{36} - 4 z_{9} z_{23} + 8 z z_{33} + 4 z_{15} z_{20} - 4 z_{6} z_{38} + 4 z_{23} z_{36} + 4 z_{28} z_{31} + 4 z_{22} z_{38} + 8 z_{9} z_{52} + 8 z_{26} z_{39} - 4 z_{29} z_{36} - 4 z_{24} z_{42} - 8 z_{22} z_{47} - 8 z_{34} z_{37} + 4 z_{30} z_{43} - 4 z_{36} z_{44} + 4 z_{31} z_{50}= 0, \\ &2 z_{15} - 4 z_{37} + 4 z_{1} z_{33} + 4 z z_{40} - 4 z_{6} z_{39} + 4 z_{23} z_{37} + 4 z_{28} z_{33} + 4 z_{24} z_{38} + 4 z_{9} z_{54} + 4 z_{27} z_{39} - 4 z_{30} z_{36} + 4 z_{15} z_{52} - 4 z_{24} z_{45} - 4 z_{22} z_{48} - 4 z_{24} z_{47} - 4 z_{35} z_{37} + 4 z_{26} z_{49} + 4 z_{30} z_{46} - 4 z_{37} z_{44} + 4 z_{33} z_{50} - 4 z_{34} z_{51}= 0, \\ &4 z_{5} z_{15} - 5 z_{51} + 8 z_{1} z_{40} - 4 z_{6} z_{49} + 8 z_{24} z_{39} - 8 z_{30} z_{37} + 4 z_{28} z_{40} + 8 z_{15} z_{54} - 4 z_{22} z_{49} - 8 z_{24} z_{48} + 4 z_{23} z_{51} + 8 z_{27} z_{49} - 4 z_{24} z_{53} + 4 z_{29} z_{51} - 8 z_{35} z_{51} + 4 z_{30} z_{58} + 4 z_{40} z_{50} - 4 z_{44} z_{51}= 0, \\ &9 z_{31} - 12 z_{9} z_{38} + 12 z_{20} z_{33} - 12 z_{23} z_{31} + 12 z_{29} z_{31} - 12 z_{37} z_{42} + 12 z_{39} z_{43} + 12 z_{31} z_{52} - 12 z_{36} z_{47}= 0, \\ &6 z_{33} - 8 z_{9} z_{39} - 4 z_{15} z_{38} - 4 z_{23} z_{33} + 4 z_{20} z_{40} + 4 z_{30} z_{31} + 8 z_{29} z_{33} - 4 z_{36} z_{39} + 4 z_{37} z_{38} - 8 z_{37} z_{45} - 4 z_{36} z_{48} - 8 z_{37} z_{47} + 4 z_{31} z_{54} + 8 z_{33} z_{52} + 8 z_{39} z_{46} + 4 z_{43} z_{49} - 4 z_{42} z_{51}= 0, \\ &3 z_{40} + 4 z_{5} z_{33} - 8 z_{15} z_{39} - 4 z_{9} z_{49} + 8 z_{30} z_{33} + 4 z_{29} z_{40} - 4 z_{36} z_{49} - 8 z_{37} z_{48} + 8 z_{33} z_{54} + 4 z_{38} z_{51} - 4 z_{37} z_{53} + 4 z_{40} z_{52} + 8 z_{46} z_{49} - 8 z_{45} z_{51} + 4 z_{39} z_{58} - 4 z_{47} z_{51}= 0, \\ &12 z_{5} z_{40} - 12 z_{15} z_{49} + 12 z_{30} z_{40} - 12 z_{37} z_{49} + 12 z_{39} z_{51} + 12 z_{40} z_{54} - 12 z_{48} z_{51} - 12 z_{51} z_{53} + 12 z_{49} z_{58}= 0, \\ &2 z_{3} z_{4} - 2 z_{2} z_{8} - z_{4} z_{11} + z_{7} z_{8} - z_{4} z_{18} + z_{10} z_{12} - z_{8} z_{19} + z_{12} z_{17} + 2 z_{12} z_{41}= 0, \\ &2 z_{2} z_{8} - 2 z_{3} z_{4} + z_{4} z_{11} - z_{7} z_{8} + z_{4} z_{18} - z_{10} z_{12} + z_{8} z_{19} - z_{12} z_{17} - 2 z_{12} z_{41}=0. \end{align} I tried to use MuPAD in MATLAB to solve this part of the equations. But MuPAD was not able to solve. Are there some other methods to solve this system of equations? Thank you very much.

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It seems this is a system of quadratic equations, so try to write it as an equation $z^T M z = 0$ with $z = (z_0,...,z_{\text{whatever}})$ and a symmetric matrix $M$ (it's always possible to choose $M$ symmetric). If the linear terms aren't typos, you can include them via $z = (1,z_0,...,z_{\text{whatever}})$.

As $M$ is symmetric, you can use some knowledge about rayleigh-quotients. E.g., you might want to check whether $M$ is positive (semi-)definite etc. I think, in order to solve $z^T M z = 0$ for non-zero $z$, you either want $M$ to have the eigenvalue $0$ and then find $z$ via $Mz = 0$ or want to express $z$ as a sum of two eigenvectors, one to a positive, one to a negative eigenvalue (so hope that $M$ is not positive/negative definite, but this can be decided by computer-algebra systems by calculation of a lot of determinants). In the end, there remains of course the question whether or not your solution $z$ is $0$ at the first component (which you want to scale to $1$).

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    $\begingroup$ I think I should correct the answer: You obviously have more than one equation, hence you could try to rewrite them as $z^TM_i z = 0$ with a lot of matrices $M_i$ and try to understand these matrices. E.g. look for a common kernel, common eigenvectors etc. $\endgroup$ – Matthias Oct 17 '15 at 19:01

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