Sums of Powers in Large Finite Fields: A Mix of Methods

Vitaly Bergelson, Andrew Best, Alex Iosevich

Type: Article

Publication Date: 2021-09-14

Citations: 2

DOI: https://doi.org/10.1080/00029890.2021.1943117

Abstract

Can any element in a sufficiently large finite field be represented as a sum of two dth powers in the field? In this article, we recount some of the history of this problem, touching on cyclotomy, Fermat's last theorem, and diagonal equations. Then, we offer two proofs, one new and elementary, and the other more classical, based on Fourier analysis and an application of a nontrivial estimate from the theory of finite fields. In context and juxtaposition, each will have its merits.

Locations

American Mathematical Monthly - View - PDF
arXiv (Cornell University) - View - PDF

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Works Cited by This (16)

Action	Title	Year	Authors
+	Theorie der algebraischen Functionen einer Veränderlichen.	1882	H. Wéber Richard Dedekind
+	Book Review: Lehrbuch der Algebra	1928	L. E. Dickson
+ PDF Chat	Characters over Certain Types of Rings with Applications to the Theory of Equations in a Finite Field	1949	L. K. Hua H. S. Vandiver
+ PDF Chat	On the Nature of the Solutions of Certain Equations in a Finite Field	1949	L. K. Hua H. S. Vandiver
+ PDF Chat	Numbers of solutions of equations in finite fields	1949	André Weil
+ PDF Chat	Sums of powers in large finite fields	1977	Charles Small
+	On Some Exponential Sums	1948	André Weil
+	Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen.	1935	H. Davenport Helmut Hasse
+	Ternary Quadratic Forms and Congruences	1926	L. E. Dickson
+	On the congruence x<sup>n</sup> + y<sup>n</sup> + z<sup>n</sup>≡0 (mod p).	1909	Liam Dickson