System of polynomial equations with a known root

Question

I have 5 polynomial equations for 5 variables and I know that the set of roots is finite. All coefficients are integers. Ultimately I'd like to find all roots but finding the Groebner basis is impossible it seems (I tried maple, mathematica, sympy) probably because it takes too long or too much memory. But I found a rather simple root just by looking at the equations :) Now I'd like to use the fact that 1 root is known to somehow reduce the set of polynomials to lower order ones, if possible, in order to simplify the problem. How would I go about this?

Answer 1

If $p = (a_1,\dotsc,a_n)$ is a known solution and your system of equations is given by the ideal $I$, then a system of equations for all the other solutions is given as follows. Let $m_p = (x_1-a_1,\dotsc,x_n-a_n)$ be the maximal ideal corresponding to $p$. Then the saturation $I:m_p^\infty$ given by $$ I:m_p^{\infty} = \{ f : \exists n, f m_p^n \subseteq I \} $$ is an ideal whose solutions (vanishing points) are precisely the solutions of $I$ other than $p$. This is under the assumption that $p$ is an isolated solution of $I$, which is the case since you know $I$ has finitely many solutions. See https://en.wikipedia.org/wiki/Ideal_quotient.

The saturation can be computed by Gröbner basis methods. For example, Macaulay2 has a command to compute the saturation. Just a guess here, but my feeling is that the saturation is easier to compute than finding the solutions other than $p$; now, whether $I:m_p^{\infty}$ is easier to solve, enough to justify that extra saturation computation, is more of a question. It probably depends on your particular ideal.

Perhaps a better approach is via numerical algebraic geometry, represented by software such as Bertini, PHCpack, etc. These software packages are exactly designed to find solutions of polynomial systems like yours. Your system is square (5 variables, 5 equations) and has finitely many solutions. If you're okay with numerical solutions, then Bertini or one of the others might be able to solve it for you. If you need "exact" solutions, then with the numerical solutions in hand, you could try something like LLL to recover "exact" solutions.