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

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 . © 2009 Elsevier Ltd. All rights reserved.