Irreducible Algebraic Sets Research Articles

Description of Coordinate Groups of Irreducible Algebraic Sets over Free 2-Nilpotent Groups

A convenient pure algebraic description of the coordinate groups of irreducible algebraic sets over a non-Abelian free 2-nilpotent group N of finite rank is given. Note that, in algebraic geometry over an arbitrary group N, it is natural to consider groups containing N as a subgroup (so-called N-groups) and homomorphisms of N-groups which are identical on N (N-homomorphisms). As a corollary, we describe all finitely generated groups H that are universally equivalent to N (with constants from N in the language). Additionally, we give a pure algebraic criterion determining when a finitely generated N-group H that is N-separated by N is, in fact, N-discriminated by N.

Coordinate groups of irreducible algebraic sets over divisible metabelian $r$-groups

Coordinate groups of irreducible algebraic sets over divisible metabelian $r$-groups

Coordinate Groups of Irreducible Algebraic Sets Over Divisible Metabelian r-Groups

Coordinate Groups of Irreducible Algebraic Sets Over Divisible Metabelian r-Groups

On algebraic geometry over completely simple semigroups

We study equations over completely simple semigroups and describe the coordinate semigroups of irreducible algebraic sets for such semigroups.

On Nash images of Euclidean spaces

In this work we characterize the subsets of Rn that are images of Nash maps f:Rm→Rn. We prove Shiota's conjecture and show that a subsetS⊂Rnis the image of a Nash mapf:Rm→Rnif and only ifSis semialgebraic, pure dimensional of dimensiond≤mand there exists an analytic pathα:[0,1]→Swhose image meets all the connected components of the set of regular points ofS. Two remarkable consequences are the following: (1) pure dimensional irreducible semialgebraic sets of dimension d with arc-symmetric closure are Nash images of Rd; and (2) semialgebraic sets are projections of irreducible algebraic sets whose connected components are Nash diffeomorphic to Euclidean spaces.

Algebraic Sets in A Divisible 2-Rigid Group

We answer the following question: Which finite unions of special irreducible algebraic sets in a divisible 2-rigid group are algebraic?

Partially Divisible Completions of Rigid Metabelian Pro-p-groups

Previously, the author defined the concept of a rigid (abstract) group. By analogy, a metabelian pro-p-group G is said to be rigid if it contains a normal series of the form G = G1 ≥ G2 ≥ G3 = 1 such that the factor group A = G/G2 is torsion-free Abelian, and G2 being a ZpA-module is torsion-free. An abstract rigid group can be completed and made divisible. Here we do something similar for finitely generated rigid metabelian pro-p-groups. In so doing, we need to exit the class of pro-p-groups, since even the completion of a torsion-free nontrivial Abelian pro-p-group is not a pro-p-group. In order to not complicate the situation, we do not complete a first factor, i.e., the group A. Indeed, A is simply structured: it is isomorphic to a direct sum of copies of Zp. A second factor, i.e., the group G2, is completed to a vector space over a field of fractions of a ring ZpA, in which case the field and the space are endowed with suitable topologies. The main result is giving a description of coordinate groups of irreducible algebraic sets over such a partially divisible topological group.

Unification and extension of intersection algorithms in numerical algebraic geometry

The solution set of a system of polynomial equations, called an algebraic set, can be decomposed into finitely many irreducible components. In numerical algebraic geometry, irreducible algebraic sets are represented by witness sets, whereas general algebraic sets allow a numerical irreducible decomposition comprising a collection of witness sets, one for each irreducible component. We denote the solution set of any system of polynomials f:CN→Cn as V(f)⊂CN. Given a witness set for some algebraic set Z⊂CN and a system of polynomials f:CN→Cn, the algorithms of this paper compute a numerical irreducible decomposition of the set Z∩V(f). While extending the types of intersection problems that can be solved via numerical algebraic geometry, this approach is also a unification of two existing algorithms: the diagonal intersection algorithm and the homotopy membership test. The new approach includes as a special case the “extension problem” where one wishes to intersect an irreducible component A of V(g(x)) with V(f(x,y)), where f introduces new variables, y. For example, this problem arises in computing the singularities of A when the singularity conditions are expressed in terms of new variables associated to the tangent space of A. Several examples are included to demonstrate the effectiveness of our approach applied in a variety of scenarios.

On irreducible algebraic sets over linearly ordered semilattices

AbstractEquations over linearly ordered semilattices are studied. For any equation

A Problem on Universal Modules

The main result in this article gives a partial answer to the following question for n = 2. Let k[U] and k[V] be the coordinate rings of the irreducible algebraic sets U and V, respectively. Is projective dimension of J n (k[U × V]) finite whenever projective dimension of J n (k[U]) and projective dimension of J n (k[V]) are finite?

Elements of Algebraic Geometry Over a Free Semilattice

It is proved that every consistent system of equations over a free semilattice of arbitrary rank is equivalent to its finite subsystem. Furthermore, irreducible algebraic sets are studied, and we look at the consistency problem for systems of equations over free semilattices.

On the number of points of algebraic sets over finite fields

We determine upper bounds on the number of rational points of an affine or projective algebraic set defined over an extension of a finite field by a system of polynomial equations, including the case where the algebraic set is not defined over the finite field by itself. A special attention is given to irreducible but not absolutely irreducible algebraic sets, which satisfy better bounds. We study the case of complete intersections, for which we give a decomposition, coarser than the decomposition in irreducible components, but more directly related to the polynomials defining the algebraic set. We describe families of algebraic sets having the maximum number of rational points in the affine case, and a large number of points in the projective case.

Numerically intersecting algebraic varieties via witness sets

The fundamental construct of numerical algebraic geometry is the representation of an irreducible algebraic set, A, by a witness set, which consists of a polynomial system, F, for which A is an irreducible component of V(F), a generic linear space L of complementary dimension to A, and a numerical approximation to the set of witness points, L∩A. Given F, methods exist for computing a numerical irreducible decomposition, which consists of a collection of witness sets, one for each irreducible component of V(F). This paper concerns the more refined question of finding a numerical irreducible decomposition of the intersection A∩B of two irreducible algebraic sets, A and B, given a witness set for each. An existing algorithm, the diagonal homotopy, computes witness point supersets for A∩B, but this does not complete the numerical irreducible decomposition. In this paper, we use the theory of isosingular sets to complete the process of computing the numerical irreducible decomposition of the intersection.

Limits of relatively hyperbolic groups and Lyndon’s completions

We describe finitely generated groups H universally equivalent (with constants from G in the language) to a given torsion-free relatively hyperbolic group G with free abelian parabolics. It turns out that, as in the free group case, the group H embeds into the Lyndon's completion G^{\mathbb{Z}[t]} of the group G , or, equivalently, H embeds into a group obtained from G by finitely many extensions of centralizers. Conversely, every subgroup of G^{\mathbb{Z}[t]} containing G is universally equivalent to G . Since finitely generated groups universally equivalent to G are precisely the finitely generated groups discriminated by G , the result above gives a description of finitely generated groups discriminated by G . Moreover, these groups are exactly the coordinate groups of irreducible algebraic sets over G .

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.

Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups

AbstractThe first main result of the paper is a criterion for a partially commutative group 𝔾 to be a domain. It allows us to reduce the study of algebraic sets over 𝔾 to the study of irreducible algebraic sets, and reduce the elementary theory of 𝔾 (of a coordinate group over 𝔾) to the elementary theories of the direct factors of 𝔾 (to the elementary theory of coordinate groups of irreducible algebraic sets).Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group ℍ. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of ℍ has quantifier elimination and that arbitrary first-order formulas lift from ℍ to ℍ * F, where F is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.

Irreducible algebraic sets over divisible decomposed rigid groups

A soluble group G is said to be rigid if it contains a normal series of the form G = G 1 > G 2 > …> G p > G p+1 = 1, whose quotients G i /G i+1 are Abelian and are torsion-free when treated as right ℤ[G/G i ]-modules. Free soluble groups are important examples of rigid groups. A rigid group G is divisible if elements of a quotient G i /G i+1 are divisible by nonzero elements of a ring ℤ[G/G i ], or, in other words, G i /G i+1 is a vector space over a division ring Q(G/G i ) of quotients of that ring. A rigid group G is decomposed if it splits into a semidirect product A 1 A 2…A p of Abelian groups A i ≅ G i /G i+1. A decomposed divisible rigid group is uniquely defined by cardinalities α i of bases of suitable vector spaces A i , and we denote it by M(α1,…, α p ). The concept of a rigid group appeared in [arXiv:0808.2932v1 [math.GR], http://xxx.lanl.gov:80/abs/0808.2932v1 ], where the dimension theory is constructed for algebraic geometry over finitely generated rigid groups. In [11], all rigid groups were proved to be equationally Noetherian, and it was stated that any rigid group is embedded in a suitable decomposed divisible rigid group M(α1,…, α p ). Our present goal is to derive important information directly about algebraic geometry over M(α1,… α p ). Namely, irreducible algebraic sets are characterized in the language of coordinate groups of these sets, and we describe groups that are universally equivalent over M(α1,…, α p ) using the language of equations.

Equations and Complexity for the Dubois–Efroymson Dimension Theorem

AbstractLetRbe a real closed field, letX⊂Rnbe an irreducible real algebraic set and letZbe an algebraic subset ofXof codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset ofXof codimension 1 containingZ. We improve this dimension theorem as follows. Indicate by μ the minimum integer such that the ideal of polynomials inR[x1, … ,xn] vanishing onZcan be generated by polynomials of degree ≤ μ. We prove the following two results: (1) There exists a polynomialP∈R[x1, … ,xn] of degree≤ μ+1 such thatX∩P–1(0) is an irreducible algebraic subset ofXof codimension 1 containingZ. (2) LetFbe a polynomial inR[x1, … ,xn] of degreedvanishing onZ. Suppose there exists a nonsingular pointxofXsuch thatF(x) = 0 and the differential atxof the restriction ofFtoXis nonzero. Then there exists a polynomialG∈R[x1, … ,xn] of degree ≤ max﹛d, μ + 1﹜ such that, for eacht∈ (–1, 1) \ ﹛0﹜, the set ﹛x∈X|F(x) +tG(x) = 0﹜ is an irreducible algebraic subset ofXof codimension 1 containingZ. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

Algebraic geometry over free metabelian lie algebras. II. Finite-field case

This paper is the second in a series of three, the object of which is to construct an algebraic geometry over the free metabelian Lie algebra F. For the universal closure of a free metabelian Lie algebra of finite rank r ⩾ 2 over a finite field k we find convenient sets of axioms in two distinct languages: with constants and without them. We give a description of the structure of finitely generated algebras from the universal closure of F r in both languages mentioned and the structure of irreducible algebraic sets over F r and respective coordinate algebras. We also prove that the universal theory of free metabelian Lie algebras over a finite field is decidable in both languages.

Irreducible Algebraic Sets in Metabelian Groups

We present the construction for a u-product G1 ○ G2 of two u-groups G1 and G2, and prove that G1 ○ G2 is also a u-group and that every u-group, which contains G1 and G2 as subgroups and is generated by these, is a homomorphic image of G1 ○ G2. It is stated that if G is a u-group then the coordinate group of an affine space Gn is equal to G ○ Fn, where Fn is a free metabelian group of rank n. Irreducible algebraic sets in G are treated for the case where G is a free metabelian group or wreath product of two free Abelian groups of finite ranks.