Multidegrees of monomial rational maps

TL;DR

This paper derives a formula for multidegrees of monomial rational maps on projective varieties, connecting them to integrals over Newton regions and volumes of polytopes, extending classical results like Bernstein-Kouchnirenko.

Contribution

It introduces a new formula for multidegrees of generalized monomial maps, linking them to geometric and combinatorial data, broadening the scope of classical theorems.

Findings

Multidegrees expressed as integrals over Newton regions.

Multidegrees related to volumes of polytopes.

A condition for computing multidegrees via characteristic polynomials.

Abstract

We prove a formula for the multidegrees of a rational map defined by generalized monomials on a projective variety, in terms of integrals over an associated Newton region. This formula leads to an expression of the multidegrees as volumes of related polytopes, in the spirit of the classical Bernstein-Kouchnirenko theorem, but extending the scope of these formulas to more general monomial maps. We also determine a condition under which the multidegrees may be computed in terms of the characteristic polynomial of an associated matrix.