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

In this article we focus on the study of special parabolic points in surfaces arising as graphs of polynomials, we prove that Viro’s construction glues a class of special parabolic points that we call transversal and build families of real polynomials in two variables with a prescribed number of special parabolic points in their graphs. When 13≤ d≤ 10,000, 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) which is the best known up until now.