Rationality of Four-Valued Families of Weil Sums of Binomials
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| Format: | Preprint |
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2023
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| author | Katz, Daniel J. Wong, Allison E. |
| author_facet | Katz, Daniel J. Wong, Allison E. |
| contents | We investigate the rationality of Weil sums of binomials of the form $W^{K,s}_u=\sum_{x \in K} ψ(x^s - u x)$, where $K$ is a finite field whose canonical additive character is $ψ$, and where $u$ is an element of $K^{\times}$ and $s$ is a positive integer relatively prime to $|K^\times|$, so that $x \mapsto x^s$ is a permutation of $K$. The Weil spectrum for $K$ and $s$, which is the family of values $W^{K,s}_u$ as $u$ runs through $K^\times$, is of interest in arithmetic geometry and in several information-theoretic applications. The Weil spectrum always contains at least three distinct values if $s$ is nondegenerate (i.e., if $s$ is not a power of $p$ modulo $|K^\times|$, where $p$ is the characteristic of $K$). It is already known that if the Weil spectrum contains precisely three distinct values, then they must all be rational integers. We show that if the Weil spectrum contains precisely four distinct values, then they must all be rational integers, with the sole exception of the case where $|K|=5$ and $s \equiv 3 \pmod{4}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_14414 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rationality of Four-Valued Families of Weil Sums of Binomials Katz, Daniel J. Wong, Allison E. Number Theory Cryptography and Security Information Theory Combinatorics 11T24, 11L05, 11L40, 11T22, 11G25, 11T71, 94A55, 94A60, 94B15 We investigate the rationality of Weil sums of binomials of the form $W^{K,s}_u=\sum_{x \in K} ψ(x^s - u x)$, where $K$ is a finite field whose canonical additive character is $ψ$, and where $u$ is an element of $K^{\times}$ and $s$ is a positive integer relatively prime to $|K^\times|$, so that $x \mapsto x^s$ is a permutation of $K$. The Weil spectrum for $K$ and $s$, which is the family of values $W^{K,s}_u$ as $u$ runs through $K^\times$, is of interest in arithmetic geometry and in several information-theoretic applications. The Weil spectrum always contains at least three distinct values if $s$ is nondegenerate (i.e., if $s$ is not a power of $p$ modulo $|K^\times|$, where $p$ is the characteristic of $K$). It is already known that if the Weil spectrum contains precisely three distinct values, then they must all be rational integers. We show that if the Weil spectrum contains precisely four distinct values, then they must all be rational integers, with the sole exception of the case where $|K|=5$ and $s \equiv 3 \pmod{4}$. |
| title | Rationality of Four-Valued Families of Weil Sums of Binomials |
| topic | Number Theory Cryptography and Security Information Theory Combinatorics 11T24, 11L05, 11L40, 11T22, 11G25, 11T71, 94A55, 94A60, 94B15 |
| url | https://arxiv.org/abs/2306.14414 |