Identities arising from Gauss sums for finite classical groups

0 Fourier Transform over Finite Field and Identities between Gauss Sums

Pure Gauss Sums over Finite Fields

New classes of pairs e,p are presented for which the Gauss sums corresponding to characters of order e over finite fields of characteristic p are pure, i.e., have a real power. Certain pure Gauss sums are explicitly evaluated. §

Bounds of Gauss Sums in Finite Fields

We consider Gauss sums of the form Gn(a) = ∑ x∈Fpm χ(x) with a nontrivial additive character χ 6= χ0 of a finite field Fpm of pm elements of characteristic p. The classical bound |Gn(a)| ≤ (n−1)pm/2 becomes trivial for n ≥ pm/2 + 1. We show that, combining some recent bounds of HeathBrown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on |Gn(a)| which is...

Identities for the Classical Polynomials Through Sums of Liouville Type

Polynomials defined recursively over the integers such as Dickson polynomials, Chebychev polynomials, Fibonacci polynomials, Lucas polynomials, Bernoulli polynomials, Euler polynomials, and many others have been extensively studied in the past. Most of these polynomials have some type of relationship between them and share a large number of interesting properties. They have been also found to b...

Infinite groups arising from partial presentations of finite groups

If G is a finite group, X a generating set for G, and R some set of relations holding among these generators (but not in general a full set of relations presenting G), we establish sufficient conditions for the group G∗ = X ! R to be infinite, and give a lower bound for the rank of the abelianization of the kernel of the natural map G∗ → G. Related results in the literature and possible directi...