We will define Newton polygons for polynomials over a valued field and prove a couple theorems using them. For example, relating the valuations of the roots of the polynomial to the slopes of the Newton polygon and proving the algebraic closure of the Puiseux series in characteristic 0.
It is known that non-negative homogeneous polynomials(forms) over $\mathbb{R}$ are same as sums of squares if it is bivariate, quadratic forms, or ternary quartic by Hilbert. Once we know a form is a sum of squares, next natural question would be how many forms are needed to represent it as sums of squares. We denote the minimal number of summands in the sums of squares by rank (of the sum of squares). Ranks of some class of forms are known. For example, any bivariate forms (allowing all monomials) can be written as sum of $2$ squares.(i.e. its rank is $2$) and every nonnegative ternary quartic can be written as a sum of $3$ squares.(i.e. its rank is $3$). Our question is that "if we do not allow some monomials in a bivariate form, how its rank will be?". In the talk, we will introduce this problem in algebraic geometry flavor and provide some notions and tools to deal with.
Milnor K-theory is a field invariant that originated as an attempt to study algebraic K-theory. Instead, Milnor K-theory has proved to have many other applications, including Galois cohomology computations, Voevodsky's proof of the Bloch-Kato conjecture, and Kato's higher class field theory. In this talk, we will go over the basic definitions and theorems of Milnor K-theory. We will also discuss some of these applications.
