Virtual Roots of Real Polynomials

TL;DR

This paper introduces 'virtual roots' as real-valued substitutes for missing roots of real polynomials, preserving sign properties and extending existing roots, with applications demonstrated.

Contribution

It proposes a novel concept of virtual roots that retain sign structure and are defined via continuous semialgebraic functions, extending the understanding of polynomial roots.

Findings

Virtual roots preserve sign properties of real roots.

Virtual roots are defined by continuous semialgebraic functions.

Applications demonstrate practical use of virtual roots.

Abstract

The fact that a real univariate polynomial misses some real roots is usually overcame by considering complex roots, but the price to pay for, is a complete lost of the sign structure that a set of real roots is endowed with (mutual position on the line, signs of the derivatives, etc...). In this paper we present real substitutes for these missing roots which keep sign properties and which extend of course the existing roots. Moreover these "virtual roots" are the values of semialgebraic continuous -- rather uniformly -- functions defined on the set of monic polynomials. We present some applications.