Certified Hermite Matrices from Approximate Roots - Univariate Case

Abstract

Let \(f_1, \ldots , f_m\) be univariate polynomials with rational coefficients and \(\mathcal {I}:=\langle f_1, \ldots , f_m\rangle \subset {\mathbb Q}[x]\) be the ideal they generate. Assume that we are given approximations \(\{z_1, \ldots , z_k\}\subset \mathbb {Q}[i]\) for the common roots \(\{\xi _1, \ldots , \xi _k\}=V(\mathcal {I})\subseteq {\mathbb C}\). In this study, we describe a symbolic-numeric algorithm to construct a rational matrix, called Hermite matrix, from the approximate roots \(\{z_1, \ldots , z_k\}\) and certify that this matrix is the true Hermite matrix corresponding to the roots \(V({\mathcal I})\). Applications of Hermite matrices include counting and locating real roots of the polynomials and certifying their existence.