Discrete Invariants of Generically Inconsistent Systems of Laurent Polynomials

2017/03/19 by Leonid Monin, Monin, Leonid
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AG #math.CO

paper · pdf · doi:10.48550/arxiv.1703.06392

10 pages, subsection 4.1 is added, comments are welcome

openalex publication_date 2017/03/19 · arxiv created 2018/09/29 · arxiv updated 2018/10/02 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Let A₁, …, A_k be finite sets in ℤⁿ and let Y ⊂ (ℂ^*)ⁿ be an algebraic variety defined by a system of equations f₁ = … = f_k = 0, where f₁, …, f_k are Laurent polynomials with supports in A₁, …, A_k. Assuming that f₁, …, f_k are sufficiently generic, the Newton polyhedron theory computes discrete invariants of Y in terms of the Newton polyhedra of f₁, …, f_k . It may appear that the generic system with fixed supports A₁, …, A_k is inconsistent. In this paper, we compute discrete invariants of algebraic varieties defined by system of equations which are generic in the set of consistent system with support in A₁, …, A_k by reducing the question to the Newton polyhedra theory. Unlike the classical situation, not only the Newton polyhedra of f₁,…,f_k, but also the supports A₁, …, A_k themselves appear in the answers.

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