Irreducible representations of semisimple algebraic groups and supersolvable lattices

Let Λ be the cross section lattice of an irreducible representation of a semisimple algebraic group. Certain combinatorial properties of Λ are studied. Supersolvable Λ?s are determined in terms of Dynkin diagrams. The characteristic polynomial of a supersolvable Λ is computed.     

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\p ^ℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall. Received: 28 May 1998 / Revised version: 20 December 1998     

Equations and algebraic geometry over profinite groups

The notion of an equation over a profinite group is defined, as well as the concepts of an algebraic set and of a coordinate group. We show how to represent the coordinate group as a projective limit of coordinate groups of finite groups. It is proved that if the set π(G) of prime divisors of the profinite period of a group G is infinite, then such a group is not Noetherian, even with respect to one-variable equations. For the case of Abelian groups, the finiteness of a set π(G) gives rise to equational Noetherianness. The concept of a standard linear pro-p-group is introduced, and we prove that such is always equationally Noetherian. As a consequence, it is stated that free nilpotent pro-p-groups and free metabelian pro-p-groups are equationally Noetherian. In addition, two examples of equationally non-Noetherian pro-p-groups are constructed. The concepts of a universal formula and of a universal theory over a profinite group are defined. For equationally Noetherian profinite groups, coordinate groups of irreducible algebraic sets are described using the language of universal theories and the notion of discriminability.     

Structure of coordinate groups for algebraic sets in partially commutative nilpotent groups

The results obtained deal in algebraic geometry over partially commutative class two nilpotent ℚ-groups, where ℚ is a field of rationals. It is proved that two arbitrary non-Abelian partially commutative class two nilpotent ℚ-groups are geometrically equivalent. A necessary and sufficient condition of being universally geometrically equivalent is specified for two partially commutative class two nilpotent ℚ-groups. Algebraic sets for systems of equations in one variable, as well as for some special systems in several variables, are described. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Translated from Algebra i Logika, Vol. 48, No. 2, pp. 378–399, May–June, 2009.     

Satisfiability of algebraic circuits over sets of natural numbers

We investigate the complexity of satisfiability problems for {∪,∩,⁻,+,×}-circuits computing sets of natural numbers. These problems are a natural generalization of membership problems for expressions and circuits studied by Stockmeyer and Meyer (1973) [10] and McKenzie and Wagner (2003) [8].Our work shows that satisfiability problems capture a wide range of complexity classes such as NL, P, NP, PSPACE, and beyond. We show that in several cases, satisfiability problems are harder than membership problems. In particular, we prove that testing satisfiability for {∩,+,×}-circuits is already undecidable. In contrast to this, the satisfiability for {∪,+,×}-circuits is decidable in PSPACE.     

Irreducible representations of locally polycyclic groups over an absolute field

Irreducible representations of locally almost polycyclic groups over an absoltue field are studied. Necessary and sufficient conditions for a solvable, locally polycyclic group of finite special rank to have a faithful irreducible representation over an absolute field are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1389–1394, October, 1990.     

Irreducible algebraic integers in short intervals

We study distribution of irreducible algebraic integers in short intervals and prove that if the class number of an algebraic number field K exceeds 2, every interval of the form (x, x + x ^θ) with a fixed θ > 1/2 contains absolute value of the norm of an irreducible algebraic integer from K provided x ≥ x ₀. The constant x ₀ effectively depends on K and θ. The author was supported in part by the grant no. N N201 1482 33 from the Polish Ministry of Science and Higher Education.     

Graphs generated by Sidon sets and algebraic equations over finite fields

We study the spectra of several graphs generated by Sidon sets and algebraic equations over finite fields. These graphs are used to study some combinatorial problems in finite fields, such as sum product estimates, solvability of some equations and the distribution of their solutions.     

Rank-1 quotient divisible groups

An Abelian group is called quotient divisible if it does not contain nonzero torsion divisible subgroups but does contain a free finite-rank subgroup such that the quotient group by it is divisible. In this paper, we will describe rank-1 quotient divisible groups with the help of cocharacteristics, and we will describe the endomorphisms of these groups as well. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 25–33, 2007.     

A class of groups in which all unconditionally closed sets are algebraic

It is proved that in any subgroup of a direct product of countable groups the property of being an unconditionally closed set coincides with that of being an algebraic set. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 8, pp. 217–222, 2006.     

Infinitely divisible measures onp-adic groups

We show that any infinitely divisible measure on ap-adic algebraic group (p a prime) has a translatex, by an elementx centralizing the support of , such thatx can be embedded in a continuous real one-parameter semigroup {V _t } _t >0, asx=v ₁.     

Structure of groups of rational points of classical algebraic groups over number fields

Dedicated to the memory of A. I. Mal'tsev.