System of polynomial equations

A system of polynomial equations (sometimes simply a polynomial system) is a set of <a href="/facts/Simultaneous_equations/KzP6TJ4u">simultaneous equations</a> f1 = 0, ..., fh = 0 where the fi are <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> in several variables, say x1, ..., xn, over some <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> k.
A solution of a polynomial system is a set of values for the xis which belong to some <a href="/facts/Algebraically_closed/HuVQClok">algebraically closed</a> <a href="/facts/Field_extension/L7yIDtyK">field extension</a> K of k, and make all equations true. When k is the field of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>, K is generally assumed to be the field of <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, because each solution belongs to a <a href="/facts/Field_extension/L7yIDtyK">field extension</a> 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 <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> and <a href="/facts/Finite_field/UkMGJo3m">finite fields</a>.
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 <a href="/facts/Diophantine_equation/mYaHp7oX">Diophantine equation</a>.

System of polynomial equations open-in-new

System of polynomial equations