A sharp bound on the number of real intersection points of a sparse plane curve with a line

We prove that the number of real intersection points of a real line with a real plane curve defined by a polynomial with at most t monomials is either infinite or does not exceed 6t−7. This improves a result by M. Avendaño. Furthermore, we prove that this bound is sharp for t=3 with the help of Grothendieck's dessins d'enfant.