Implicit function theorem over free groups

We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as Implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ systems. We will also show that the Implicit function theorem is true only for these varieties. In algebraic geometry such results would be described as lifting solutions of equations into generic points. From the model theoretic view-point we claim the existence of simple Skolem functions for particular $\forall\exists$-formulas over free groups. Proving these theorems we describe in details a new version of the Makanin-Razborov process for solving equations in free groups. We also prove a weak version of the Implicit function theorem for NTQ systems which is one of the key results in the solution of the Tarski's problems about the elementary theory of a free group.Comment: 144 pages, 16 figure

Irreducible Affine Varieties over a Free Group II. Systems in Triangular Quasi-quadratic Form and Description of Residually Free Groups

AbstractWe shall prove the conjecture of Myasnikov and Remeslennikov [4] which states that a finitely generated group is fully residually free (every finite set of nontrivial elements has nontrivial images under some homomorphism into a free group) if and only if it is embeddable in the Lyndon's exponential groupFZ[x], which is theZ[x]-completion of the free group. HereZ[x] is the ring of polynomials of one variable with integer coefficients. Historically, Lyndon's attempts to solve Tarski's famous problem concerning the elementary equivalence of free groups of different ranks led him to introduceFZ[x].An ∃-free group is a groupGsuch that the class of ∃-formulas, true inG, is the same as the class of ∃-formulas, true in a nonabelian free group. A finitely generated group is ∃-free if and only if it is fully residually free [22]. Our result gives an algebraic description of ∃-free groups.We shall give an algorithm to represent a solution set of an arbitrary system of equations overFas a union of finite number of irreducible components in the Zariski topology onFn. The solution set for every system is contained in the solution set of a finite number of systems in triangular form with quadratic words as leading terms. The possibility of such a decomposition for a solution set was conjectured by Razborov in [20] and also by Rips.We shall give a description of systems of equations determining irreducible components using methods developed in [13,19]; it is possible to find some of these methods in [18]. We are thankful to E. Rips for attracting our attention to these techniques

Finding All Solutions of Equations in Free Groups and Monoids with Involution

The aim of this paper is to present a PSPACE algorithm which yields a finite graph of exponential size and which describes the set of all solutions of equations in free groups as well as the set of all solutions of equations in free monoids with involution in the presence of rational constraints. This became possible due to the recently invented emph{recompression} technique of the second author. He successfully applied the recompression technique for pure word equations without involution or rational constraints. In particular, his method could not be used as a black box for free groups (even without rational constraints). Actually, the presence of an involution (inverse elements) and rational constraints complicates the situation and some additional analysis is necessary. Still, the recompression technique is general enough to accommodate both extensions. In the end, it simplifies proofs that solving word equations is in PSPACE (Plandowski 1999) and the corresponding result for equations in free groups with rational constraints (Diekert, Hagenah and Gutierrez 2001). As a byproduct we obtain a direct proof that it is decidable in PSPACE whether or not the solution set is finite.Comment: A preliminary version of this paper was presented as an invited talk at CSR 2014 in Moscow, June 7 - 11, 201

More Than 1700 Years of Word Equations

Geometry and Diophantine equations have been ever-present in mathematics. Diophantus of Alexandria was born in the 3rd century (as far as we know), but a systematic mathematical study of word equations began only in the 20th century. So, the title of the present article does not seem to be justified at all. However, a linear Diophantine equation can be viewed as a special case of a system of word equations over a unary alphabet, and, more importantly, a word equation can be viewed as a special case of a Diophantine equation. Hence, the problem WordEquations: "Is a given word equation solvable?" is intimately related to Hilbert's 10th problem on the solvability of Diophantine equations. This became clear to the Russian school of mathematics at the latest in the mid 1960s, after which a systematic study of that relation began. Here, we review some recent developments which led to an amazingly simple decision procedure for WordEquations, and to the description of the set of all solutions as an EDT0L language.Comment: The paper will appear as an invited address in the LNCS proceedings of CAI 2015, Stuttgart, Germany, September 1 - 4, 201

Minsky machines and algorithmic problems

This is a survey of using Minsky machines to study algorithmic problems in semigroups, groups and other algebraic systems.Comment: 19 page