1999/11/14 by Teresa Krick, Térésa Krick, Krick, Teresa +5 · 2 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Primary: 11G35 #Secondary: 13P10 #math.AC #math.AG #math.NT #msc:11G35 #msc:13P10
paper · pdf · doi:10.48550/arxiv.math/9911094
55 pages, LaTeX2e
arxiv created 1999/11/14 · openalex publication_date 1999/11/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over \Z. The result improves previous work of Philippon, Berenstein-Yger and Krick-Pardo. We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system. As an application, we derive an effective arithmetic Nullstellensatz for sparse polynomial systems. The proof of these results relies heavily on the notion of local height of an affine variety defined over a number field. We introduce this notion and study its basic properties.