System of polynomial equations

A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations $f 1 = 0, ..., f h = 0$ where the $f i$ are polynomials in several variables, say $x 1, ..., x n$ , over some field $k$ .

A solution of a polynomial system is a set of values for the $x i$ s which belong to some algebraically closed field extension $K$ of $k$ , and make all equations true. When $k$ is the field of rational numbers, $K$ is generally assumed to be the field of complex numbers, because each solution belongs to a field extension of $k$ , which is isomorphic to a subfield of the complex numbers.

This article is about the methods for solving, that is, finding all solutions or describing them. As these methods are designed for being implemented in a computer, emphasis is given on fields $k$ in which computation (including equality testing) is easy and efficient, that is the field of rational numbers and finite fields.

Searching for solutions that belong to a specific set is a problem which is generally much more difficult, and is outside the scope of this article, except for the case of the solutions in a given finite field. For the case of solutions of which all components are integers or rational numbers, see Diophantine equation.