2009/01/22 by Martín Avendaño, Avendano, Martin, Ashraf Ibrahim +1
Mathematics · #14Q99 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.0901.3393
openalex publication_date 2009/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be a complete non-archimedean field with a discrete valuation, f∈ K[X] a polynomial with non-vanishing discriminant, A the valuation ring of K, and \M the maximal ideal of A. The first main result of this paper is a reformulation of Hensel's lemma that connects the number of roots of f with the number of roots of its reduction modulo a power of \M. We then define a condition --- \em regularity --- that yields a simple method to compute the exact number of roots of f in K. In particular, we show that regularity implies that the number of roots of f equals the sum of the numbers of roots of certain binomials derived from the Newton polygon.