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We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the…
Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of…
We study the distribution of singular and unimodular matrices in sumsets in matrix rings over finite fields. We apply these results to estimate the largest prime divisor of the determinants in sumsets in matrix rings over the integers.
We consider a class of optimization problems that involve determining the maximum value that a function in a particular class can attain subject to a collection of difference constraints. We show that a particular linear programming…
In the theory of approximation there are some problems on approximation of compacts in functional spaces by nonlinear families : first we deal with the polynomial case, and then we consider the analytic case. We demonstrate a negative…
We introduce and study the problem of optimizing arbitrary functions over degree sequences of hypergraphs and multihypergraphs. We show that over multihypergraphs the problem can be solved in polynomial time. For hypergraphs, we show that…
We present an intimate connection among the following fields: (a) distributed local algorithms: coming from the area of computer science, (b) finitary factors of iid processes: coming from the area of analysis of randomized processes, (c)…
The zeta function of a motive over a finite field is multiplicative with respect to the direct sum of motives. It has beautiful analytic properties, as were predicted by the Weil conjectures. There is also a multiplicative zeta function,…
We discuss a conjecture concerning the enumeration of nonsingular matrices over a finite field that are block companion and whose order is the maximum possible in the corresponding general linear group. A special case is proved using some…
As predictive algorithms grow in popularity, using the same dataset to both train and test a new model has become routine across research, policy, and industry. Sample-splitting attains valid inference on model properties by using separate…
We reinterpret a conjecture of Breuil on the locally analytic $\mathrm{Ext}^1$ in a functorial way using $(\varphi,\Gamma)$-modules (possibly with $t$-torsion) over the Robba ring, making it more accurate. Then we prove several special or…
We briefly introduce several problems: (1) a generalization of the convex fair partition conjecture, (2) on non-trivial invariants among polyhedrons that can be formed from the same set of face polygons, (3) two questions on assembling…
We investigate the behaviour of a certain additive function depending on prime divisors of specific integers lying in large gaps between consecutive primes. The result is obtained by a combination of results and ideas related to large gaps…
Sampling algorithms, hypergraph degree sequences, and polytopes play a crucial role in statistical analysis of network data. This article offers a brief overview of open problems in this area of discrete mathematics from the point of view…
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…
Interestingly, LLMs yet struggle with some basic tasks that humans find trivial to handle, e.g., counting the number of character r's in the word "strawberry". There are several popular conjectures (e.g., tokenization, architecture and…
Let $K$ be a number field. We show that, up to allowing a finite set of denominators in the partial quotients, it is possible to define algorithms for $\mathfrak P$-adic continued fractions satisfying the finiteness property on $K$ for…
In this paper an iterated function system on the space of distribution functions is built. The inverse problem is introduced and studied by convex optimization problems. Some applications of this method to approximation of distribution…
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…
Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of `Idempotent Mathematics' with an emphasis on matrix theory,…