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I am interested in solving binomial systems of the form $$ \begin{cases} a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} + b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\ \vdots &\vdots \\ a_m x_1^{d_{m1}} x_2^{d_{m2}} \cdots x_n^{d_{mn}} + b_m x_1^{d_{m1}} x_2^{d_{m2}} \cdots x_n^{d_{mn}} &= 0 \end{cases} $$ where the exponents may be negative. I.e., each equation has exactly two terms. For example $$ \begin{cases} 3 x_1^{2} x_2^{-5} + 4 x_1^{-1} x_2^{6} &= 0 \\ 2 x_1^{3} x_2^{5} - 7 x_1^{2} x_2^{4} &= 0 \end{cases} $$ I am looking for some software packages that can solve these systems the nonzero complex numbers. Here is what I mean by "solve":

  • When the solution set consists of isolated points. Locate the points.
  • When the solution set has positive dimension, find the number of the components and the dimension. Ideally we should also get the parametrization of each component.
  • Multiplicity information would be nice but not necessary.

Binomials.m2, a Maclaulay2 package ( seems to be the best. But when applied to large systems with 50 or more variables, Binomials.m2 simply does not terminate within any reasonable amount of time.

I am familiar with the algorithms behind, but before I write my own, I would like to know if there is anything even better out there. Or maybe I'm using Binomials.m2 incorrectly?

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