Year: 2009
Type: preprint
Abstract: Let $K$ be a complete non-archimedean field with a discrete valuation, $f\in K[X]$ 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.
Source: arXiv (Cornell University)
Authors: Avendano, Martin, Ibrahim, Ashraf
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1666030848