On Newton Diagrams of Plurisubharmonic Polynomials

TL;DR

This paper investigates the geometric properties of Newton diagrams of plurisubharmonic polynomials in two complex variables, demonstrating that certain unions of extreme edges do not necessarily produce plurisubharmonic polynomials.

Contribution

It constructs a specific example of a plurisubharmonic polynomial with two extreme edges whose union or convex hull does not yield a plurisubharmonic polynomial, challenging previous assumptions.

Findings

Union of extreme edges may not produce plurisubharmonic polynomials.

Convex hull of union of extreme edges may not be plurisubharmonic.

Provides a counterexample to a common belief in the geometry of Newton diagrams.

Abstract

Each extreme edge of the Newton diagram of a plurisubharmonic polynomial on $\mathbb{C}^2$ gives rise to a plurisubharmonic polynomial. It is tempting to believe that the union of the extreme edges or the convex hull of said union will do the same. We construct a plurisubharmonic polynomial $P$ on $\mathbb{C}^2$ with precisely two extreme edges $E_1$ and $E_2$, such that neither $E_1\cup{E_2}$ nor $\text{Conv}({E_1\cup{}E_2})$ yields a plurisubharmonic polynomial.