On sums of powers of consecutive squares over finite fields, and sums of distinct values of polynomials

TL;DR

This paper calculates sums of powers over finite fields for elements with specific quadratic properties and applies these results to determine sums over polynomial image sets, resolving two conjectures.

Contribution

It provides explicit formulas for sums of powers of elements with quadratic constraints and solves conjectures about sums over polynomial images in finite fields.

Findings

Explicit formulas for sums of powers of quadratic elements

Resolved two conjectures on sums over polynomial images

Characterized sum behavior for specific polynomial forms

Abstract

For each odd prime power q, and each integer k, we determine the sum of the k-th powers of all elements x in F_q for which both x and x+1 are squares in F_q^*. We also solve the analogous problem when one or both of x and x+1 is a nonsquare. We use these results to determine the sum of the elements of the image set f(F_q) for each f(X) in F_q[X] of the form X^4+aX^2+b, which resolves two conjectures by Finch-Smith, Harrington, and Wong.