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Let a finite collection of (complex) unknowns $\{x_1,\ldots,x_n\}$ be given, as well as an affine system $AX=B$ in the quadratic variables $X:=[x_i x_j : i\leq j]$, with entries in a computable subfield of $\mathbb C$.

Is there a way of deciding whether the system has a solution? If the answer is positive, is there an efficient way, without resorting to general Gröbner basis algorithms (which in general fail to solve the systems I have at hands, e.g. Maple and Sympy implementations) to solve it?

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    $\begingroup$ This is decidable (e.g., by quantifier elimination for algebraically closed fields). However, you can reduce the satisfiability of arbitrary polynomial systems to quadratic systems by introducing extra variables; so solutions certainly do not lie in a tower of quadratic extensions. $\endgroup$
    – Emil Jeřábek
    Commented Jun 14, 2022 at 12:12
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    $\begingroup$ Thanks Emil, I removed the last part of my post. Also, I gather that solving quadratic systems is as difficult as solving general ones, hence there probably is no specific algorithms out there. $\endgroup$
    – Loïc Teyssier
    Commented Jun 14, 2022 at 12:15

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