Power sums of polynomials over finite fields and applications: a survey. (English) Zbl 1378.11101
Summary: 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.
MSC:
| 11T55 | Arithmetic theory of polynomial rings over finite fields |
| 11-02 | Research exposition (monographs, survey articles) pertaining to number theory |
| 11G09 | Drinfel’d modules; higher-dimensional motives, etc. |
| 11R58 | Arithmetic theory of algebraic function fields |
| 11R60 | Cyclotomic function fields (class groups, Bernoulli objects, etc.) |
| 11M32 | Multiple Dirichlet series and zeta functions and multizeta values |
| 11M38 | Zeta and \(L\)-functions in characteristic \(p\) |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 14H40 | Jacobians, Prym varieties |
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