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A function field analog of Jacobi’s theorem on sums of squares and its moments

Part of: Finite fields and commutative rings (number-theoretic aspects) Multiplicative number theory

Published online by Cambridge University Press:  10 January 2025

Wentang Kuo ,
Yu-Ru Liu  and
Yash Totani
Wentang Kuo
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L3G1, Canada e-mail: [email protected] [email protected]
Yu-Ru Liu
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L3G1, Canada e-mail: [email protected] [email protected]
Yash Totani*
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L3G1, Canada e-mail: [email protected] [email protected]
*
e-mail: [email protected]
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Abstract

In this article, we establish a function field analog of Jacobi’s theorem on sums of squares and analyze its moments. Our approach involves employing two distinct techniques to derive the main results concerning asymptotic formulas for the moments. The first technique utilizes Dirichlet series framework to derive asymptotic formulas in the limit of large finite fields, specifically when the characteristic of $\mathbb {F}_q[T]$ becomes large. The second technique involves effectively partitioning the set of polynomials of a fixed degree, providing asymptotic formulas in the limit of large polynomial degree.

Keywords

Sums of squaresquadratic formsJacobi’s theoremnumber of representationspolynomial rings

MSC classification

Primary: 11T55: Arithmetic theory of polynomial rings over finite fields 11N37: Asymptotic results on arithmetic functions
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 17
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

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