Power Sum Varieties over Finite Fields

Source abstract

We study a family of affine algebraic varieties over a finite field $\mathbb{F}_q$ defined by systems of power sum polynomials. Our main goal is to investigate their geometric and arithmetic properties, with particular emphasis on the structure and dimension of their singular loci. Using tools from the theory of symmetric polynomials and generalized Vandermonde determinants, we establish explicit conditions under which these varieties are complete intersections and prove that, in a broad range of parameters, their singular loci have codimension at least two. As a consequence, we derive normality and irreducibility results for both the affine varieties and their projective closures. Building on this geometric analysis, we obtain estimates for the number of $\mathbb{F}_q$-rational points and deduce effective existence results for solutions of systems of symmetric power sum equations over finite fields. Finally, we apply these results to the study of Waring's problem and to the analysis of the cost of computing a $\mathbb{F}_q$-rational point of such systems.