The number of roots of a lacunary bivariate polynomial on a line

Martín Avendaño

doi:10.1016/j.jsc.2008.02.016

Abstract

We prove that a polynomial f ∈ R [ x , y ] with t non-zero terms, restricted to a real line y = a x + b , either has at most 6 t − 4 zeros or vanishes over the whole line. As a consequence, we derive an alternative algorithm for deciding whether a linear polynomial y − a x − b ∈ K [ x , y ] divides a lacunary polynomial f ∈ K [ x , y ] , where K is a real number field. The number of bit operations performed by the algorithm is polynomial in the number of non-zero terms of f , in the logarithm of the degree of f , in the degree of the extension K / Q and in the logarithmic height of a , b and f .

Highlights

The famous Descartes’ rule of signs, 1641, establishes that the number of positive real roots of a polynomial f ∈ R[x], counted with multiplicities, is bounded by the number of changes of signs in its ordered vector of coefficients, skipping the zeros
The number of real roots of f is bounded by 2t − 1, where t is its number of non-zero terms
A simple version of it implies that a square system of n real polynomial equations in n indeterminates, involving a total t non-zero terms, has at most (n + 1)t 2t(t−1)/2 non-degenerate roots in the positive orthant

Summary

Introduction

The famous Descartes’ rule of signs, 1641, establishes that the number of positive real roots of a polynomial f ∈ R[x], counted with multiplicities, is bounded by the number of changes of signs in its ordered vector of coefficients, skipping the zeros. The number of real roots of f is bounded by 2t − 1, where t is its number of non-zero terms (here all roots are counted with multiplicities, except 0 which is counted at most once). In Li et al (2003), Li, Rojas and Wang (see Perruci (2005)) studied particular cases of bivariate square systems and showed that the number of common isolated or non-degenerate roots of a trinomial and a polynomial with at most t non-zero terms, t ≥ 3, is bounded above by 2t − 2. The first algorithm for this purpose can be deduced from a more general result by Kaltofen and Koiran (2005) They presented a polynomial-time algorithm for computing all linear factors of a lacunary bivariate polynomial. We reduce the problem to the univariate case by considering specializations f (x, xn) for small values of n

Changes of signs

This means that f