Numerical irreducible decomposition over a number field

Abstract

Let [Formula: see text] be a set of elements in the polynomial ring [Formula: see text], let [Formula: see text] denote the ideal generated by the elements of [Formula: see text], and let [Formula: see text] denote the radical of [Formula: see text]. There is a unique decomposition [Formula: see text] with each [Formula: see text] a prime ideal corresponding to a minimal associated prime of [Formula: see text] over [Formula: see text]. Let [Formula: see text] denote the reduced algebraic set corresponding to the common zeroes of the elements of [Formula: see text]. Techniques from numerical algebraic geometry can be used to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text]. This corresponds to producing a witness set for [Formula: see text] for each [Formula: see text] together with the degree and dimension of [Formula: see text] (a point in a witness set for [Formula: see text] can be considered as a numerical approximation for a general point on [Formula: see text]). The purpose of this paper is to show how to extend these results taking into account the field of definition for the polynomial system. In particular, let [Formula: see text] be a number field (i.e. a finite field extension of [Formula: see text]) and let [Formula: see text] be a set of elements in [Formula: see text]. We show how to extend techniques from numerical algebraic geometry to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text].