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

Abstract

We prove that a bivariate polynomial f with exactly t non-zero terms, restricted to a real line {y=ax+b}, either has at most 6t-4 zeroes or vanishes over the whole line. As a consequence, we derive an alternative algorithm to decide whether a linear polynomial divides a bivariate polynomial (with exactly t non-zero terms) over a real number field K within [ log(H(f)H(a)H(b)) [K:Q}] log(deg(f)) t]^{O(1)} bit operations.