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Let $G$ be a finite group of order $n$, and $\xi$ an $n$-th primitive root of unity. Consider the affine scheme $C:=\mbox{Spc}({\mathbb Z}[\xi]\otimes_{\mathbb Z} R(G))$ where $R(G)$ is the representation ring of $G$. We study the fibers of…
We classify finite groups that can act by automorphisms and birational automorphisms on non-trivial Severi-Brauer surfaces over fields of characteristic zero.
The goal and the main result of the paper is to provide a complete description of the field of rational differential invariants of one class of second order ordinary differential equations with scalar control parameter with respect to Lie…
The special linear representation of a compact Lie group G is a kind of linear representation of compact Lie group G with special properties. It is possible to define the integral of linear representation and extend this concept to special…
It is shown that a trivial version of polarization is sufficient to produce separating systems of polynomial invariants: if two points in the direct sum of the $G$--modules $W$ and $m$ copies of $V$ can be separated by polynomial…
We construct a birational invariant for certain algebraic group actions. We use this invariant to classify linear representations of finite abelian groups up to birational equivalence, thus answering, in a special case, a question of E. B.…
In this paper, we define locally matchable subsets of a group which is extracted from the concept of matchings in groups and used as a tool to give alternative proofs for existing results in matching theory. We also give the linear analogue…
We obtain a new classification of the finite metacyclic group in terms of group invariants. We present an algorithm to compute these invariants, and hence to decide if two given finite metacyclic groups are isomorphic, and another algorithm…
We study differential forms invariant under a finite reflection group over a field of arbitrary characteristic. In particular, we prove an analogue of Saito's freeness criterion for invariant differential 1-forms. We also discuss how…
We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we…
The spectra of a finite group is the set of its element orders. We obtain an arithmetic description of finite symplectic and orthogonal groups. In particular, a description of spectra of all finite simple simplectic and orthogonal groups is…
This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…
Inspired by methods in prime characteristic in commutative algebra, we introduce and study combinatorial invariants of seminormal monoids. We relate such numbers with the singularities and homological invariants of the semigroup ring…
We estimate, in a number field, the number of elements and the maximal number of linearly independent elements, with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices.…
This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups…
The grouplike elements of a coalgebra over a field are known to be linearly independent over said field. Here we prove three variants of this result. One is a generalization to coalgebras over a commutative ring (in which case the linear…
The problem of representation of elements of weighted space of infinitely differentiable functions on real line by exponential series is considered.
We associate with a matrix over an arbitrary field an infinite family of matrices whose sizes vary from one to infinity; their entries are traces of powers of the original matrix. We explicitly evaluate the determinants of matrices in our…
We call a group $G$ {\it algorithmically finite} if no algorithm can produce an infinite set of pairwise distinct elements of $G$. We construct examples of recursively presented infinite algorithmically finite groups and study their…
A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the…