Related papers: Sum of two squares in biquadratic fields

This is the second in a series of three papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We give sharp conditions on the entries of a positive semidefinite NxN matrix function F on n-dimensional…

The general formulas for finding the quantity of all primitive and nonprimitive triples generated by the given number x have been proposed. Also the formulas for finding the complete quantity of the representations of the integers as a…

Let K = Q(\omega) with \omega^3 = m be a pure cubic number field. We show that the elements\alpha \in K^\times whose squares have the form a - \omega form a group isomorphic to the group of rational points on the elliptic curve E_m: y^2=…

Let $p$ and $q$ be polynomials with degree $2$ over an arbitrary field $\mathbb{F}$. In the first part of this article, we characterize the matrices that can be decomposed as $A+B$ for some pair $(A,B)$ of square matrices such that $p(A)=0$…

Asymptotic formulae for Titchmarsh-type divisor sums are obtained with strong error terms that are uniform in the shift parameter. This applies to more general arithmetic functions such as sums of two squares, improving the error term in…

We settle an old question about the existence of certain "sums-of-squares" formulas over a field F (which are the simplest examples of composition formulas for quadratic forms). A classical theorem says that if such a formula exists over a…

We address the problem of when two finite dimensional central division algebras over the same field are necessarily isomorphic given that they have the same maximal subfields.

There is a classical geometric construction which uses a binary quadratic form to define an involution on the space of binary d-ics. We give a complete characterization of a general class of such involutions which are definable using…

We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which…

When the Euclidean algorithm produces a symmetric sequence of quotients, we give explicit formulas for the remainders that allow the analysis of two families of quadratic forms in the remainders.

For a family of polytopes of even dimension $2p$, known as \textit{dual-neighborly}, it has been shown for $p\ne 2$ that the associated intersection of quadrics is a connected sum of sphere products $S^p\times S^p$. In this article we give…

This paper develops an approach to the evaluation of quadratic Euler sums that involve harmonic numbers. The approach is based on simple integral computations of polyloga- rithms. By using the approach, we establish some relations between…

The sum-product algorithm for decoding of binary codes is analyzed for bipartite graphs in which the check nodes all have degree $2$. The algorithm simplifies dramatically and may be expressed using linear algebra. Exact results about the…

In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for…

This paper is a survey of author's mathematical and logical study of the problem of quantization of fields.

We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms…

We consider the problem of finding 4 rational squares, such that the product of any two plus the sum of the same two always gives a square. We give some historical background and exhibit one such quadruple.

This paper is concerned with the inverse scattering problem for the three-dimensional Maxwell's equations in bi-anisotropic periodic structures. The inverse scattering problem aims to determine the shape of bi-anisotropic periodic…

Unifying several directions of the development of the study of summing multilinear operators between Banach spaces, we construct a general framework that studies, under one single definition, multilinear operators that are summing with…

We give an efficient method of counting bi-colored A'Campo forests and provide a stratification criterion for the space of those graphs. We deduce an algorithm to count these forests in polynomial time, answering an open question of…