2007/03/23 by Monica del Pilar Canales, Mónica del Pilar Canales, Canales, Monica del Pilar
Computer Science · Engineering · Mathematics · #11P05 #11T22 #11T23 #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #graph theory and CDMA systems #math.NT #msc:11P05 #msc:11T22 #msc:11T23 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0703707
12 pages
arxiv created 2007/03/23 · openalex publication_date 2007/03/23 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let sd(p,a) = min \k | a = ∑i=1kaid, ai∈ \ffp^*\ be the smallest number of d-th powers in the finite field Fp, sufficient to represent the number a in Fp^*. Then gd(p) = maxa in Fp^* sd(p,a) gives an answer to Waring's Problem mod p. We first introduce cyclotomic integers n(k,ν), which then allow to state and solve Waring's problem mod p in terms of only the cyclotomic numbers (i,j) of order d. We generalize the reciprocal of the Gaussian period equation G(T) to a C-differentiable function I(T) in Q[[T]], which also satisfies I'(T)/I(T) in Z[[T]]. We show that and why a≡ -1 mod Fp*d (the classical "Stufe", if d = 2) behaves special: Here (and only here) I(T) is in fact a polynomial from Z[T], the reciprocal of the period polynomial. We finish with explicit calculations of gd(p) for the cases d = 3 and d = 4, all primes p, using the known cyclotomic numbers compiled by Dickson.