This paper introduces two new results on Kloosterman sums and their minimal polynomials. We characterise ternary Kloosterman sums modulo 27. We also prove a congruence concerning the minimal polynomial over Q of a Kloosterman sum. This paper also serves as a survey of our recent results on binary Kloosterman sums modulo 16, 32, 64 and 128 with Petr Lisoněk.

Lehmer proved that the values of the cosine function evaluated at rational multiples of π are algebraic numbers. We show how to determine explicit, closed form expressions for the minimal polynomials of these algebraic numbers.

We develop a method of creating skew congruences on subpowers of finite algebras using groups of twin polynomials, and apply it to the investigation of residually small varieties generated by nilpotent algebras. We prove that a residually small variety generated by a finite nilpotent (in particular, a solvable Eminimal) algebra is weakly abelian. Conversely, we show in two special cases that a ...

Recently, Fukuda and Komatsu constructed units of a certain abelian extension of Q(exp(2π √ −1/5)) using special values of Siegel modular functions. In this paper, we determine the minimal polynomials of these units.

We present several generalisations of the GamesChan algorithm. For a fixed monic irreducible polynomial f we consider the sequences s that have as characteristic polynomial a power of f . We propose an algorithm for computing the linear complexity of s given a full (not necessarily minimal) period of s. We give versions of the algorithm for fields of characteristic 2 and for arbitrary finite ch...

We present several generalisations of the GamesChan algorithm. For a fixed monic irreducible polynomial f we consider the sequences s that have as characteristic polynomial a power of f . We propose an algorithm for computing the linear complexity of s given a full (not necessarily minimal) period of s. We give versions of the algorithm for fields of characteristic 2 and for arbitrary finite ch...

We address the concept of “minimal polynomial encoder” for finite support linear convolutional codes over Zpr . These codes can be interpreted as polynomial modules which enables us to apply results from the 2007-paper [8] to introduce the notions of “p-encoder” and “minimal p-encoder”. Here the latter notion is the ring analogon of a row reduced polynomial encoder from the field case. We show ...

For P 2 Z[x], let kPk denote the Euclidean norm of the coe cient vector of P . For an algebraic number , with minimal polynomial A, de ne the Euclidean norm of by k k = kkAk ; where k is the smallest positive integer for which kA 2 Z[x]. De ne the minimal Euclidean norm of by k k min = min kPk : P 2Z[x]; P ( ) = 0; P 6 0 : Given an algebraic number , we show there exists a P 2 Z[x] with P ( ) =...

Let E be an elliptic curve defined over Q. The aim of this paper is to make it possible to compute Heegner L-functions and anticyclotomic Λ-adic regulators of E, which were studied by Mazur-Rubin and Howard. We generalize results of Cohen and Watkins and thereby compute Heegner points of nonfundamental discriminant. We then prove a relationship between the denominator of a point of E defined ov...