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.

T. A. Akoglu—partially supported by TUBITAK grant 119F211.

A. Szanto—partially supported by NSF grants CCF-1813340 and CCF-1217557.