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

TL;DR

This paper develops a Viro's patchworking method to construct real polynomial graphs with a large number of special parabolic points, approaching the known upper bounds for degree d polynomials.

Contribution

It introduces a theorem of Viro's patchworking type for creating polynomial graphs with a prescribed number of special parabolic points, advancing the understanding of their maximum possible count.

Findings

Constructed polynomials with $(d-4)(2d-9)$ special parabolic points

Achieved counts closer to the upper bound $(d-2)(5d-12)$ for degree d

Enhanced methods for controlling parabolic point distribution in polynomial graphs

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 $d \geq 13$, which is the best known up until now.