Absolute irreducibility is a fundamental property of algebraic varieties: a variety is absolutely irreducible if it remains irreducible after extending the base field to its algebraic closure. Determining absolute irreducibility is an important problem in algebraic geometry, particularly over finite fields, where it has applications to the study of rational points, coding theory, cryptography, and finite geometry. In particular, absolute irreducibility plays an important role in establishing bounds for the number of rational points on varieties and in the study of exceptional functions, such as almost perfect nonlinear (APN) and permutation functions. In this talk, I will discuss criteria for establishing the absolute irreducibility of hypersurfaces defined by multivariate polynomials over finite fields. I will present a criterion that, under a square-free hypothesis on the leading homogeneous component, reduces the problem to computations involving multivariate greatest common divisors and avoids the need to test irreducibility over the ground field or its extensions. Since square-free polynomials constitute a large and natural class, this provides an effective method for proving absolute irreducibility in many situations. I will also discuss applications of these techniques to polynomial families arising from problems over finite fields.

We consider a system of elliptic equations, depending on a small parameter, that models long-range segregation of populations. The diffusion is governed by the Laplacian. This system was previously investigated by Caffarelli, Patrizi, and Quitalo as a model in population dynamics, and they established the regularity of the free boundary in dimension two. In the present work we study the free boundary in the higher dimensional case. We extend the concept of angles and asymptotic cones to higher dimensions, and give a characterization of regular and singular points in terms of their densities and angles. We obtain a structure result of the free boundary and show that, if the angles at the singular points are away from $\frac{n\omega_n}{2}$, the regular set is open in the free boundary and locally a $C^1$ manifold of dimension $n-1$. We also show that, if the supports of the populations are convex, they are convex polytopes. A weak form of the equality of angles for the convex configuration is also derived. The talk is based on a joint work with Professor Monica Torres.

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