We investigate the linear complexity and the minimal polynomial over the finite fields of the characteristic sequences of cubic and biquadratic residue classes. Also we find the linear complexity and the minimal polynomial of the balanced cyclotomic sequences of order three. Keywords—linear complexity, finite field, cubic residue classes, biquadratic residue classes

We present a new conjecture relating the minimal polynomial solution of the level-one U q (sl(2)) quantum Knizhnik-Zamolodchikov equation for generic values of q in the link pattern basis and some q-enumeration of Totally Symmetric Self-Complementary Plane Partitions.

Metric heights are modified height functions on the non-zero algebraic numbers Q ∗ which can be used to define a metric on certain cosets of Q ∗ . They have been defined with a view to eventually applying geometric methods to the study of Q ∗ . In this paper we discuss the construction of metric heights in general. More specifically, we study in some detail the metric height obtained from the n...

Let V be a vector space over some field k, and let α : V V be a linear map (an ‘endomorphism of V ’). Given any polynomial p with coefficients in k, there is an endomorphism p(α) of V , and we say that p is an annihilating polynomial for α if p(α) = 0. Our first major goal is to see that for any α, the annihilating polynomials can easily be classified: they’re precisely the multiples of a certa...

We will develop the machinery of resultants and resolvent polynomials with the ultimate goal of understanding the so-called “resolvent method” for computing Galois groups over Q. Along the way, we will encounter tangential applications of these tools, such as the determination of the minimal polynomial of, say, the sum of two algebraic numbers using resultants. In this way, this paper can serve...

This article considers the n-number Ducci game as a linear map over the n dimensional vector space ZZn 2 . Period lengths are characterized as orders of factors of the minimal polynomial of that transformation. The minimal polynomial can be explicitly obtained for any n. We provide algebraic proofs of divisibility conditions, initially found by Ehrlich. In addition, all periodic cycle lengths f...

Let F, G and H be non-empty graphs. The notation F → (G,H) means that if any edge of F is colored by red or blue, then either the red subgraph of F con- tains a graph G or the blue subgraph of F contains a graph H. A graph F (without isolated vertices) is called a Ramsey (G,H)−minimal if F → (G,H) and for every e ∈ E(F), (F − e) 9 (G,H). The set of all Ramsey (G,H)−minimal graphs is denoted by ...

In this paper, after reviewing some results in minimal space, some new results in this setting are given. We prove a generalized form of the Fan-KKM typetheorem in minimal vector spaces. As some applications, the open type of matching theorem and generalized form of the classical KKM theorem in minimal vector spaces are given.

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, which enable us to compute Heegner points of non-fundamental discriminant. We then prove a relationship between the denominator of a point of E d...

We consider the sequences over the finite field of four elements obtained by inverse Gray mapping from a pair of binary sequences. We derive the linear complexity and the minimal polynomial of sequences constructed from Legendre sequences, Hall’s sextic sequences and twin-prime sequences using the technique proposed by Tang, Ding, Lim, Kim et al.