Toric Euler-Jacobi vanishing theorem and zeros at infinity

TL;DR

This paper explores conditions under which residues in toric geometry vanish globally, linking these to zeros at infinity and interpolation problems, extending classical results to more general settings.

Contribution

It provides geometric criteria on Newton polytopes that connect residue vanishing to zeros at infinity in toric varieties, generalizing Khovanskii's theorem.

Findings

Residue vanishing is equivalent to zeros at infinity under certain geometric conditions.

Conditions relate to the dimension of the quotient of the Cox ring by the input polynomials.

The results connect residue theory with interpolation problems in algebraic geometry.

Abstract

Residues appear naturally in various questions in complex and algebraic geometry: interpolation, duality, representation problems, and obstructions. The first global vanishing result in the projective plane, known as the Euler-Jacobi theorem, was established by Jacobi in 1835. In the toric case, the input is a system of n Laurent sparse polynomials with fixed Newton polytopes, and the first version of the Euler-Jacobi toric vanishing theorem for residues in the n-torus is due to Khovanskii in 1978, under restrictive genericity assumptions. In this paper, we provide geometric conditions on the input Newton polytopes to ensure that this global vanishing is equivalent to the existence of zeros at infinity in the associated compact toric variety. We relate these conditions to the dimension at the toric critical degree of the quotient of the Cox ring by the ideal generated by the (multi)homogenizations of the input polynomials. We also relate the existence of zeros at infinity to interpolation questions.