[Paper Review] Residues and tame symbols in toric geometry

Ivan Soprounov|arXiv (Cornell University)|Mar 12, 2002

Commutative Algebra and Its Applications3 references3 citations

TL;DR

This paper introduces a unified framework using Parshin’s residues and tame symbols on toroidal varieties to generalize and explain Khovanskii’s formula for the product of roots and Gelfond-Khovanskii’s formula for the sum of Grothendieck residues in systems of equations on (C×)ⁿ. The approach extends these results to algebraically closed fields of arbitrary characteristic, providing a geometric and cohomological interpretation of classical formulas in toric geometry.

ABSTRACT

Abstract. We introduce a new approach to study of systems of algebraic equations in (C × ) n whose Newton polytopes have sufficiently general relative locations. Recently A. Khovanskii found an explicit formula for the product of the roots of such a system [Kh1]. Also an explicit formula for the sum of the Grothendieck residues over the roots of such a system was found by O. Gelfond and A. Khovanskii [G-Kh]. Our approach gives a uniform explanation of both these results in terms of Parshin’s residues and tame symbols on toroidal varieties, and extends these results to the case of an algebraically closed field of

Motivation & Objective

• To provide a geometric and cohomological interpretation of classical formulas for systems of equations on (C×)ⁿ with Newton polytopes in general relative position.
• To unify Khovanskii’s formula for the product of roots and Gelfond-Khovanskii’s formula for the sum of Grothendieck residues using Parshin’s residues and tame symbols.
• To extend these results from the complex numbers to algebraically closed fields of arbitrary characteristic.
• To establish a framework in toroidal varieties that naturally accommodates both residue and symbol constructions in the context of toric geometry.

Proposed method

• Utilizing Parshin’s higher-dimensional residues on toroidal varieties to generalize classical residue theory to higher codimension cycles.
• Applying tame symbols associated with divisors in toric varieties to capture multiplicative invariants of the system’s roots.
• Employing the structure of Newton polytopes and their relative positions to define the relevant cohomological classes.
• Constructing a pairing between tame symbols and residues via duality in the cohomology of toric varieties.
• Using the geometry of toric varieties to interpret the product of roots and sum of residues as global invariants.
• Extending the formalism from the complex case to algebraically closed fields via algebraic geometry over arbitrary fields.

Experimental results

Research questions

• RQ1 How can Parshin’s residues and tame symbols be used to provide a unified explanation of Khovanskii’s product formula and Gelfond-Khovanskii’s sum formula?
• RQ2 What is the role of Newton polytopes with general relative positions in enabling a uniform residue-theoretic treatment?
• RQ3 In what way do tame symbols on toroidal varieties encode the multiplicative structure of solutions to systems of equations?
• RQ4 How can the classical formulas over ℂ be extended to algebraically closed fields of arbitrary characteristic?
• RQ5 What geometric and cohomological structures underlie the duality between residue and symbol constructions in this setting?

Key findings

• The paper provides a uniform cohomological framework that explains both Khovanskii’s product formula and Gelfond-Khovanskii’s sum formula through Parshin’s residues and tame symbols.
• The method successfully generalizes the formulas from the complex numbers to algebraically closed fields of arbitrary characteristic.
• The tame symbols on toroidal varieties are shown to encode the multiplicative invariants of the roots of the system.
• The sum of Grothendieck residues over the isolated solutions is expressed as a global residue pairing via the toric structure.
• The product of roots is interpreted as a tame symbol pairing, extending Khovanskii’s result to arbitrary algebraically closed fields.
• The framework reveals a deep duality between residue and symbol constructions in the context of toric geometry and Newton polytopes.