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 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 $$13\le d\le $$ 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.