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

We prove that a polynomial f ∈ R[x, y] with t non-zero terms, restricted on 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 y−ax−b ∈ K[x, y] divides a sparse polynomial f ∈ K[x, y] with t terms in [log(H(f)H(a)H(b))[K : Q] log(deg(f))t] bit operations, where K is a real number field.