## 群論セミナー： Newton polygons, precision and slope factorization of polynomials

2016年 　 2月 15日 14時 45分 ～ 2016年 　 2月 15日 16時 15分

Tristan Vaccon (JSPS Fellow (post-doc), 立教大学)

Let K = Q_p or K((T)) (or any complete discretely valued field).
A very classical tool for the study of polynomials in K[X] is the Newton
polygon.

Given a polynomial f in K[X], the Newton polygon of f, NP(f) is a convex
polygon in the real plane defined by the coefficients of f.
A classical application to this notion is an irreducibility criterion more
general than Eisenstein's, called Dumas's criterion.

We can use Newton polygons to follow the precision on arithmetic operations
over polynomials over K (multiplication, modular multiplication).
We compare this method with the optimal method of differential precision
(developped by Caruso-Roe-V.).

An important application of Newton polygons is the so-called slope
factorization of f: one factor of f is defined by slope of NP(f), which has
proved to be useful in the study of p-adic differential equations.
The factors in the slope factorization are prime to one another.
In a joint work with X.Caruso and D.Roe (work in progress), we define a
Newton method to obtain the factors of the slope factorization of f in K[X].