An explicit calculation of the Ronkin function

TL;DR

This paper explicitly computes the second derivatives of the Ronkin function for affine linear polynomials in three variables, expressing them through elliptic integrals and hypergeometric functions, thereby providing a semi-explicit form of the Ronkin measure.

Contribution

It introduces a precise calculation of the Ronkin function's derivatives for specific polynomials, linking them to special functions and advancing understanding of the associated Monge-Ampère measure.

Findings

Explicit formulas for second derivatives of the Ronkin function

Representation of the Ronkin measure via elliptic integrals and hypergeometric functions

Enhanced understanding of the Monge-Ampère measure in this context

Abstract

We calculate the second order derivatives of the Ronkin function in the case of an affine linear polynomial in three variables and give an expression of them in terms of complete elliptic integrals and hypergeometric functions. This gives a semi-explicit expression of the associated Monge-Amp\`ere measure, the Ronkin measure.