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

摘要

We prove that a polynomial f ∈ R[x, y] with t non-zero terms, restricted to a real line y = ax + b, either has at most 6t - 4 zeros or vanishes over the whole line. As a consequence, we derive an alternative algorithm for deciding whether a linear polynomial y - ax - 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 nonzero 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.