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I'm trying to solve a bilinear system of equations over a finite field. (More specifically: I'm trying to find a single solution, if one exists.) The system consists of equations of the form $$y^T A_i ...

Let $f_1,\ldots,f_n\in \mathbb{Q}[X_1,\ldots,X_n]$ be a system of $n$ polynomials in $n$ indeterminant, which only has finitely many solutions. Supose that the each of the variables $X_i$ appears at ...