Power sums of polynomials over finite fields and applications: A survey

In this brief expository survey, we explain some results and conjectures on various aspects of the study of the sums of integral powers of monic polynomials of a given degree over a finite field. The aspects include non-vanishing criteria, formulas and bounds for degree and valuation at finite primes, explicit formulas of various kind for the sums themselves, factorizations of such sums, generating functions for them, relations between them, special type of interpolations of the sums by algebraic functions, and the resulting connections between the motives constructed from them and the zeta and multizeta special values. We mention several applications to the function field arithmetic.