Related papers: On sesquilinear forms over finite fields

We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…

We use invariant theory of finite groups to study shape enumerators of self-dual linear codes in a finite NRT metric space. We provide a new approach that avoids applying Molien's formula to compute all possible shape enumerators. We also…

Each graph and choice of a commutative ring gives rise to an associated graphical group. In this article, we introduce and investigate graph polynomials that enumerate conjugacy classes of graphical groups over finite fields according to…

The problem of the classification of the indefinite binary quadratic forms with integer coefficients is solved introducing a special partition of the de Sitter world, where the coefficients of the forms lie, into separate domains. Every…

Primarily this paper presents an expository report on alternatives to the traditional methods of classifying representations of finite dimensional algebras. Some new results illustrating such alternatives for algebras with only finitely…

We develop Kummer theory for algebraic function fields in finitely many transcendental variables. We consider any finitely generated Kummer extension (possibly, over a cyclotomic extension) of an algebraic function field, and describe the…

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 this paper, we construct several new permutation polynomials over finite fields. First, using the linearized polynomials, we construct the permutation polynomial of the form $\sum_{i=1}^k(L_{i}(x)+\gamma_i)h_i(B(x))$ over ${\bf…

We present new classes of permutation polynomials over finite fields.

A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary…

Permutation polynomials over finite fields have taken an important role in vast areas in mathematics as well as engineering. Recently, Tu et al. gave some classes of complete permutation polynomials over finite fields of even…

Let $q$ be a power of a prime and $\mathbb{F}_q$ the finite field consisting of $q$ elements. We prove explicit upper bounds on the number of incidences between lines and Cartesian products in $\mathbb{F}_q^2$. We also use our results on…

We obtain transformation formulas for quadratic character sums with quartic and cubic polynomial arguments.

We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke…

In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite…

We construct the quasi-classical approximation of the form factors in finite volume using the separation of variables. The latter is closely related to the Baxter equation.

We study non-linear Schr\"odinger operators on graphs. We construct minimal nonnegative solutions to corresponding semi-linear elliptic equations and use them to introduce the notion of stochastic completeness at infinity in a non-linear…

We improve lower bounds on the $k$th-order nonlinear complexity of pseudorandom sequences over finite fields and we establish a probabilistic result on the behavior of the $k$th-order nonlinear complexity of random sequences over finite…

We consider an arbitrary representation of the additive group over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants.

We present an explicit formula for subregular characters (i.e, irreducible finite-dimensional complex characters of submaximal degree) of the unitriangular group over a finite field of sufficiently large characteristic.