Speaker:
Hanna Ćmiel (University of Szczecin)

Title:
The Newton polygon and values of roots of polynomials

Abstract:
Consider a polynomial in one variable over a given valued field. To such a 
polynomial one can assign a value by defining it to be the minimum among the 
values of its coefficients (this is called the Gauß valuation). In general, 
when given a polynomial, we only have knowledge about its coefficients. The 
Newton Polygon is a tool that allows us to derive the values of roots of the 
polynomial from the values of its coefficients. In this talk, we have a look 
at the Newton Polygon and its applications. We describe the construction of 
the Newton Polygon and we state the well-known theorem on the connection 
between the values of the coefficients of a polynomial and the values of its 
roots. We then state a theorem on similarities of the Newton Polygons of 
polynomials which are close to each other in the topology induced by the Gauß
valuation. Armed with those two theorems, we will then be able to give precise
statements not only on the values of roots of polynomials which are close to 
each other, but also on how close the respective roots are to each other under 
a suitable pairing.