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

Zhiguo Ding; Michael E. Zieve

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arXiv · 2309.14979

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

Zhiguo Ding·Michael E. Zieve

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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.

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Zhiguo Ding, Michael E. Zieve. 2023-09-26. On sums of powers of consecutive squares over finite fields, and sums of distinct values of polynomials. https://arxiv.org/abs/2309.14979

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