Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Toric Euler-Jacobi vanishing theorem and zeros at infinity

Carlos D'Andrea, Alicia Dickenstein

TL;DR

This work extends the Euler–Jacobi vanishing framework to toric residues, establishing conditions under which global residue vanishing in the torus corresponds to the absence of zeros at toric infinity. By introducing indecomposable polytope sequences and analyzing critical degrees, toric Jacobians, and determinants $\Delta_{\boldsymbol{F},\sigma}$, the authors link residue vanishing to the dimension of Cox-ring quotients and to local multiplicities. The paper develops a robust residue theory in toric and simplicial settings, connecting global and local residues, and applying it to interpolation questions. The results offer a geometric criterion for vanishing and provide tools for computational tests via codimension-one maps, with open questions guiding future exploration on finiteness, general indecomposability, and broader applicability.

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.

Toric Euler-Jacobi vanishing theorem and zeros at infinity

TL;DR

This work extends the Euler–Jacobi vanishing framework to toric residues, establishing conditions under which global residue vanishing in the torus corresponds to the absence of zeros at toric infinity. By introducing indecomposable polytope sequences and analyzing critical degrees, toric Jacobians, and determinants , the authors link residue vanishing to the dimension of Cox-ring quotients and to local multiplicities. The paper develops a robust residue theory in toric and simplicial settings, connecting global and local residues, and applying it to interpolation questions. The results offer a geometric criterion for vanishing and provide tools for computational tests via codimension-one maps, with open questions guiding future exploration on finiteness, general indecomposability, and broader applicability.

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.
Paper Structure (21 sections, 28 theorems, 78 equations)

This paper contains 21 sections, 28 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. A bit of history and our aim
  3. The converse of the Euler-Jacobi Theorem and other results
  4. Basics on Toric varieties
  5. Toric varieties from fans
  6. Divisors and polytopes
  7. Toric varieties from lattice polytopes
  8. The case of $n+1$ divisors
  9. Special polynomials of critical degree
  10. Codimension one
  11. Toric residues and residues in the torus
  12. Global residues
  13. Local residues and duality in the smooth case
  14. Residues in simplicial affine varieties
  15. Toric and global residues in the simplicial case
  16. ...and 6 more sections

Key Result

Theorem 1.1

( Toric Euler-Jacobi theorem ) Assume $V_T({\boldsymbol{f}})$ is finite. If the sum of the local multiplicities at all the points $\xi \in V_{\boldsymbol{f}}$ equals $MV(P_1, \dots, P_n)$ (equivalently, the closures of the hypersurfaces $(f_i=0)$ do not have any common point of intersection at the d

Theorems & Definitions (45)

  • Theorem 1.1
  • Example 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • ...and 35 more