Transversal special parabolic points in the graph of a polynomial obtained under Viro’s patchworking

Abstract

In this article we focus on the study of special parabolic points in surfaces arising as graphs of polynomials, we give a theorem of Viro's patchworking type to build families of real polynomials in two variables with a prescribed number of special parabolic points in their graphs. We use this result to build a family of degree d real polynomials in two variables with (d−4)(2d−9) special parabolic points in its graph. This brings the number of special parabolic points closer to the upper bound of (d − 2)(5d − 12) when d ≥ 13, which is the best known up until now.