Sparse effective membership problems via residue currents

2009/03/20 by Elizabeth Wulcan, Wulcan, Elizabeth
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Complex Variables (math.CV) #FOS: Mathematics #Polynomial and algebraic computation #math.AG #math.CV

paper · pdf · doi:10.48550/arxiv.0903.3618

23 pages

openalex publication_date 2009/03/20 · arxiv created 2010/08/20 · arxiv updated 2010/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use residue currents on toric varieties to obtain bounds on the degrees of solutions to polynomial ideal membership problems. Our bounds depend on (the volume of) the Newton polytope of the polynomial system and are therefore well adjusted to sparse polynomial systems. We present sparse versions of Max Nöther's AF+BG Theorem, Macaulay's Theorem, and Kollár's Effective Nullstellensatz, as well as recent results by Hickel and Andersson-Götmark.

Related

• Some variants of Macaulay's and Max Noether's theorems
• The membership problem for polynomial ideals in terms of residue currents
• On the effective membership problem for polynomial ideals
• The algebraic degree of sparse polynomial optimization
• A sparse effective Nullstellensatz