Related papers: Polyadic integer numbers and finite (m,n)-fields
Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic…
We construct an analogue of the ring of algebraic numbers, living in a quotient of the product of all finite fields of prime order. We use this ring to deduce some results about linear recurrent sequences.
This paper presents some basic theorems giving the structure of cyclic codes of length n over the ring of integers modulo p^a and over the p-adic numbers, where p is a prime not dividing n. An especially interesting example is the 2-adic…
It is stated that Boolean set algebras with unit V, where V is a union of Cartesian products, are axiomatizable. The axiomatization coincides with that of cylindric polyadic equality algebras (class CPE). This is an algebraic representation…
It is shown that the sum of class numbers of orders in totally complex quartic fields with no real quadratic subfield obeys an asymptotic law similar to the prime numbers, as the bound on the regulators tends to infinity. Here only orders…
We present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case. Our bottom line is a straightforward connection between apolarity of binary forms and the inner…
This paper introduces a novel approach to the axiomatic theory of quadratic forms. We work internally in a category of certain partially ordered sets, subject to additional conditions which amount to a strong form of local presentability.…
In characteristic zero, Zinovy Reichstein and the author generalized the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative…
It is proved that for any finite dimensional representation of a prime order group over the field of rational numbers, polynomial invariants of degree at most $3$ separate the orbits. A result providing an upper degree bound for separating…
In this short survey we look at a few basic features of p-adic numbers, somewhat with the point of view of a classical analyst. In particular, with p-adic numbers one has arithmetic operations and a norm, just as for real or complex…
We introduce axiomatically a Nonarchimedean field E, called the field of the Euclidean numbers, where a transfinite sum is defined that is indicized by ordinal numbers less than the first inaccessible {\Omega}. Thanks to this sum, E becomes…
We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. The central problem is whether and how certain rings are (additively) generated by their units.…
We present effective upper bounds on the symmetric bilinear complexity of multiplication in extensions of a base finite field Fp2 of prime square order, obtained by combining estimates on gaps between prime numbers together with an optimal…
We previously showed that the inverse limit of standard-graded polynomial rings with perfect coefficient field is a polynomial ring, in an uncountable number of variables. In this paper, we show that the same result holds with arbitrary…
In this paper, as an analogue of the integer case, we define congruence preserving functions over the residue class rings of polynomials over finite fields. We establish a counting formula for such congruence preserving functions, determine…
Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible…
We prove a function field analogue of Maynard's result about primes with restricted digits. That is, for certain ranges of parameters n and q, we prove an asymptotic formula for the number of irreducible polynomials of degree n over a…
Let $R$ be an order in an algebraic number field. If $R$ is a principal order, then many explicit results on its arithmetic are available. Among others, $R$ is half-factorial if and only if the class group of $R$ has at most two elements.…
For any given positive integer $m$, a necessary and sufficient condition for the existence of Type I $m$-adic constacyclic codes is given. Further, for any given integer $s$, a necessary and sufficient condition for $s$ to be a multiplier…
Previous research on exceptional units has primarily focused on the ring of rational integers or abstract finite rings, often restricted to linear or quadratic constraints. In this paper, we extend the concept of polynomial-type exceptional…