In this talk, we will focus the discussion on the Hilbert-Kunz functions. If R is an affine semigroup rings, then it is not difficult to see that, by the Erhart Theory, these functions are quasipolynomials, i.e., polynomial functions with periodic coefficients. We will see also how the studies of algebraic intersection theory, jointly with K. Kurano, may help us, understand the stabilities of the Hilbert-Kunz functions (not necessarily of semigroup rings). Furthermore, to create rings whose Hilbert-Kunz functions are in the form of polynomials with prescribed coefficients.
In algebraic statistics, the models of interest are algebraic sets, i.e., solution sets to a system of multivariate polynomial equations. In this situation, the number of critical points of the squared Euclidean distance function on the model's Zariski closure is a topological invariant called the Euclidean Distance (ED) Degree.
In this talk, I will present some models from computer vision and statistics that may be described as algebraic sets. Moreover, I will describe a topological method for determining Euclidean distance degree and a numerical algebraic geometry approach for determining critical points of the squared Euclidean distance function.