Ultrametric Root Counting
Martin Avendano, Ashraf Ibrahim

TL;DR
This paper introduces a reformulation of Hensel's lemma and a regularity condition that simplifies counting roots of polynomials over non-archimedean fields by relating it to roots of binomials from the Newton polygon.
Contribution
It provides a new reformulation of Hensel's lemma and introduces a regularity condition for straightforward root counting in non-archimedean fields.
Findings
Reformulation of Hensel's lemma relating roots of polynomials and their reductions.
Definition of a regularity condition for root counting.
Root count equals sum of roots of binomials from the Newton polygon.
Abstract
Let be a complete non-archimedean field with a discrete valuation, a polynomial with non-vanishing discriminant, the valuation ring of , and the maximal ideal of . The first main result of this paper is a reformulation of Hensel's lemma that connects the number of roots of with the number of roots of its reduction modulo a power of . We then define a condition --- {\em regularity} --- that yields a simple method to compute the exact number of roots of in . In particular, we show that regularity implies that the number of roots of equals the sum of the numbers of roots of certain binomials derived from the Newton polygon.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
Topicsadvanced mathematical theories · Algebraic Geometry and Number Theory · Meromorphic and Entire Functions
