2023/06/26 by Daniel J. Katz, Katz, Daniel J., Allison E. Wong +1
Computer Science · Mathematics · #11G25 #11L05 #11L40 #11T22 #11T24 #11T71 #94A55 #94A60 #94B15 #Algebraic Geometry and Number Theory #Coding theory and cryptography #Combinatorics (math.CO) #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Information Theory (cs.IT) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2306.14414
openalex publication_date 2023/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the rationality of Weil sums of binomials of the form WK,su=∑x ∈ K ψ(xs - u x), where K is a finite field whose canonical additive character is ψ, and where u is an element of K× and s is a positive integer relatively prime to |K^×|, so that x ↦ xs is a permutation of K. The Weil spectrum for K and s, which is the family of values WK,su as u runs through K^×, 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^×|, 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 ≡ 3 \pmod4.