Affine Algebraic Sets Research Articles

Recognition and Analysis of Image Patterns Based on Latin Squares by Means of Computational Algebraic Geometry

With the particular interest of being implemented in cryptography, the recognition and analysis of image patterns based on Latin squares has recently arisen as an efficient new approach for classifying partial Latin squares into isomorphism classes. This paper shows how the use of a Computer Algebra System (CAS) becomes necessary to delve into this aspect. Thus, the recognition and analysis of image patterns based on these combinatorial structures benefits from the use of computational algebraic geometry to determine whether two given partial Latin squares describe the same affine algebraic set. This paper delves into this topic by focusing on the use of a CAS to characterize when two partial Latin squares are either partial transpose or partial isotopic.

A computational algebraic geometry approach to analyze pseudo-random sequences based on Latin squares

Latin squares are used as scramblers on symmetric-key algorithms that generate pseudo-random sequences of the same length. The robustness and effectiveness of these algorithms are respectively based on the extremely large key space and the appropriate choice of the Latin square under consideration. It is also known the importance that isomorphism classes of Latin squares have to design an effective algorithm. In order to delve into this last aspect, we improve in this paper the efficiency of the known methods on computational algebraic geometry to enumerate and classify partial Latin squares. Particularly, we introduce the notion of affine algebraic set of a partial Latin square L = (lij) of order n over a field $\mathbb {K}$ as the set of zeros of the binomial ideal $\langle x_{i}x_{j}-x_{l_{ij}}\colon (i,j) \text { is a non-empty cell in} L \rangle \subseteq \mathbb {K}[x_{1},\ldots ,x_{n}]$ . Since isomorphic partial Latin squares give rise to isomorphic affine algebraic sets, every isomorphism invariant of the latter constitutes an isomorphism invariant of the former. In particular, we deal computationally with the problem of deciding whether two given partial Latin squares have either the same or isomorphic affine algebraic sets. To this end, we introduce a new pair of equivalence relations among partial Latin squares: being partial transpose and being partial isotopic.

Congruences and coordinate semirings of tropical varieties

In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence. Our main definition applies Zariski density to the algebraic structure of the coordinate semiring† of an affine supertropical algebraic set, which we tie to tropical geometry, especially in connection with the dimension of an affine variety, obtaining the analogs of classical results from dimension theory including catenarity. The second approach, based on the layered structure, is given in the appendix.

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.

Parameterized affine codes

Let K be a finite field and let X* be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Gröbner bases, to compute the length and the dimension of CX* (d), the parameterized affine code of degree d on the set X*. If Y is the projective closure of X*, it is shown that CX* (d) has the same basic parameters that CY (d), the parameterized projective code on the set Y. If X* is an affine torus, we compute the basic parameters of CX* (d). We show how to compute the vanishing ideals of X* and Y.

On theorems of Delsarte–McEliece and Chevalley–Warning–Ax–Katz

We present a theorem that generalizes the result of Delsarte and McEliece on the p-divisibilities of weights in abelian codes. Our result generalizes the Delsarte---McEliece theorem in the same sense that the theorem of N. M. Katz generalizes the theorem of Ax on the p-divisibilities of cardinalities of affine algebraic sets over finite fields. As the Delsarte---McEliece theorem implies the theorem of Ax, so our generalization implies that of N. M. Katz. The generalized theorem gives the p-divisibility of the t-wise Hamming weights of t-tuples of codewords (c (1), . . . ,c (t)) as these words range over a product of abelian codes, where the t-wise Hamming weight is defined as the number of positions i in which the codewords do not simultaneously vanish, i.e., for which $${(c^{(1)}_i,\ldots,c^{(t)}_i)\not=(0,\ldots,0)}$$ . We also present a version of the theorem that, for any list of t symbols s 1, . . . ,s t , gives p-adic estimates of the number of positions i such that $${(c^{(1)}_i,\ldots,c^{(t)}_i)=(s_1,\ldots,s_t)}$$ as these words range over a product of abelian codes.

On algebraic cellular automata

We investigate some general properties of algebraic cellular automata, i.e., cellular automata over groups whose alphabets are affine algebraic sets and which are locally defined by regular maps. When the ground field is assumed to be uncountable and algebraically closed, we prove that such cellular automata always have a closed image with respect to the prodiscrete topology on the space of configurations and that they are reversible as soon as they are bijective.

The ADHM variety and perverse coherent sheaves

We study the full set of solutions to the ADHM equation as an affine algebraic set, the ADHM variety. We determine a filtration of the ADHM variety into subvarieties according to the dimension of the stabilizing subspace. We compute dimension, and analyze singularity and reducibility of all of these varieties. We also establish a connection between arbitrary solutions of the ADHM equation and coherent perverse sheaves on P 2 in the sense of Kashiwara.

Lie algebras and algebraic groups

Preface 1. Results on topological spaces 1.1 Irreducible sets and spaces 1.2 Dimension 1.3 Noetherian spaces 1.4 Constructible sets 1.5 Gluing topological spaces 2. Rings and modules 2.1 Ideals 2.2 Prime and maximal ideals 2.3 Rings of fractions and localization 2.4 Localization of modules 2.5 Radical of an ideal 2.6 Local rings 2.7 Noetherian rings and modules 2.8 Derivations 2.9 Module of differentials 3. Integral extensions 3.1 Integral dependence 3.2 Integrally closed rings 3.3 Extensions of prime ideals 4. Factorial rings 4.1 Generalities 4.2 Unique factorization 4.3 Principal ideal domains and Euclidean domains 4.4 Polynomial and factorial rings 4.5 Symmetric polynomials 4.6 Resultant and discriminant 5. Field extensions 5.1 Extensions 5.2 Algebraic and transcendental elements 5.3 Algebraic extensions 5.4 Transcendence basis 5.5 Norm and trace 5.6 Theorem of the primitive element 5.7 Going Down Theorem 5.8 Fields and derivations 5.9 Conductor 6. Finitely generated algebras 6.1 Dimension 6.2 Noether's Normalization Theorem 6.3 Krull's Principal Ideal Theorem 6.4 Maximal ideals 6.5 Zariski topology 7. Gradings and filtrations 7.1 Graded rings and graded modules 7.2 Graded submodules 7.3 Applications 7.4 Filtrations 7.5 Grading associated to a filtration 8. Inductive limits 8.1 Generalities 8.2 Inductive systems of maps 8.3 Inductive systems of magmas, groups and rings 8.4 An example 8.5 Inductive systems of algebras 9. Sheaves of functions 9.1 Sheaves 9.2 Morphisms 9.3 Sheaf associated to a presheaf 9.4 Gluing 9.5 Ringed space 10. Jordan decomposition and some basic results on groups 10.1 Jordan decomposition 10.2 Generalities on groups 10.3 Commutators 10.4 Solvable groups 10.5 Nilpotent groups 10.6 Group actions 10.7 Generalities on representations 10.8 Examples 11. Algebraic sets 11.1 Affine algebraic sets 11.2 Zariski topology 11.3 Regular functions 11.4 Morphisms 11.5 Examples of morphisms 11.6 Abstract algebraic sets 11.7 Principal open subsets 11.8 Products of algebraic sets 12. Prevarieties and varieties 12.1 Structure sheaf 12.2 Algebraic prevarieties 12.3 Morphisms of prevarieties 12.4 Products of prevarieties 12.5 Algebraic varieties 12.6 Gluing 12.7 Rational functions 12.8 Local rings of a variety 13. Projective varieties 13.1 Projective spaces 13.2 Projective spaces and varieties 13.3 Cones and projective varieties 13.4 Complete varieties 13.5 Products 13.6 Grassmannian variety 14. Dimension 14.1 Dimension of varieties 14.2 Dimension and the number of equations 14.3 System of parameters 14.4 Counterexamples 15. Morphisms

A note on separation of algebraic sets and the Łojasiewicz exponent for polynomial mappings

In the paper a global separation problem for affine algebraic sets is considered. As application an upper bound for the distance of the graph of polynomial mapping to its zero set in the form of a Łojasiewicz inequality is given.

TUNNEL NUMBER ONE KNOTS HAVE PALINDROME PRESENTATIONS

We show that any tunnel number one knot group has a two generator one relator presentation in which the relator is a palindrome in the generators. We use this fact to compute the character variety for this knot groups and we show that it is an affine algebraic set [Formula: see text].

Affine Algebraic Varieties Relative to an Algebraic Theory

Affine algebraic varieties relative to an algebraic theory are introduced and described as irreducible components of affine algebraic sets. Their category is shown to be dually equivalent to the category of irreducible functional algebras.

Finitely coordinated and finitely copresentable affine algebraic sets

In order to describe the finite character of geometrical objects, the notions of finitely coordinated and finitely copresentable objects are introduced. It is shown that they work well in geometrical categories of affine algebraic sets. Their connections with the finitely generated and finitely presentable objects of Gabriel–Ulmer are established. The noetherian case, where finitely coordinated objects are identical to finitely copresentable ones, is studied.

Affine algebraic sets relative to an algebraic theory

For an arbitrary algebraic theory T and a prescribed T-algebraL, the geometrical category AfAlgSet(L) of affine algebraic sets over the affine lineL is build up. It is proved to be dually equivalent to the category FcAlg(L) of functional T-algebras overL. The Galois theory ofL is defined and a Galois duality established. These geometrical categories are characterized up to an equivalence of categories.

An algorithm to obtain laws of families of filiform Lie algebras

We present a polynomial time algorithm which generates families of filiform Lie algebras of dimension n. By this recursive algorithm, we obtain parametrizations of the affine algebraic sets of filiform Lie algebras of dimensions 11 and 12.

Rings of SL 2 ( $\Bbb{C}$ )-characters and the Kauffman bracket skein module

Let M be a compact orientable 3-manifold. The set of characters of SL2($\Bbb {C}$)-representations of $\pi\_1(M)$ forms a closed affine algebraic set. We show that its coordinate ring is isomorphic to a specialization of the Kauffman bracket skein module, modulo its nilradical. This is accomplished by realizing the module as a combinatorial analog of the ring in which tools of skein theory are exploited to illuminate relations among characters. We conclude with an application, proving that a small manifold's specialized module is necessarily finite dimensional.

Tor and Torsion on a Complete Intersection

Let (R,m) be a complete intersection, that is, a ring whose m-adic completion is the quotient of a regular local ring by a regular sequence. SupposeMandNare finitely generatedR-modules. We give a necessary condition for the vanishing of TorRi(M,N) for alli⪢0 in terms of the intersection of certain affine algebraic sets associated toMandN. We apply this condition to the study of torsion in tensor products. For example, we show that ifRis a domain andMis anR-module of infinite projective dimension then there exist infinitely manynfor which the tensor product ofMwith one of itsnth syzygy modules has torsion.We also give a sufficient condition for the vanishing TorRi(M,N) for alli⪢0 in terms of the ability to leftMandNto “disjoint” complete intersections of smaller codimension. We use this condition to construct tensor products of non-free modules which are maximal Cohen–Macaulay.

The Set of Types ofn-Dimensional Semisimple and Cosemisimple Hopf Algebras Is Finite

In this paper we give a partial answer to the above statement, proving that the set of types of semisimple and cosemisimple Hopf algebras of a Ž . given dimension is finite in any characteristic . The proof uses the formalism of deformation theory of bialgebras which is concerned, from a Ž . geometrical point of view, with the study of the affine algebraic set B V of all bialgebra structures which can be defined on a vector space V. Ž . There are two basic facts regarding B V . First of all, the transport of Ž . Ž . structures defines an action of GL V on B V , such that the orbit of a bialgebra A contains all structures isomorphic to A, so the set of types of bialgebras on V is in a one-to-one correspondence with the set of orbits. Second, we are led in a natural way to a cohomology theory for bialgebras, Ž . Ž Ž .. studying the tangent spaces T O and T B V , where O is the closure A A A A of the orbit through a bialgebra A. One of the most important results of the deformation theory of bialgebras is that the orbit of a bialgebra is open if the cohomology vanishes. Therefore, the number of orbits which correspond to bialgebras having trivial cohomology is finite. We obtain our result by proving that the cohomology of a semisimple Hopf algebra A vanishes if the characteristic of k is 0, or more generally, if A is

Low dimensional sections of basic semialgebraic sets

Let X be a real affine algebraic set and S a semialgebraic set. Many important results are known about the basicness of S: mainly, if S is basic open, S can be defined by s strict inequalities, where s is bounded by the dimension of X. It is also known (Br¨ocker-Scheiderer criterion) that an open semialgebraic set is basic and is defined by s inequalities (s-basic) if and only if for every irreducible subset Y of X the intersection S Y is generically s-basic. In a previous paper, the authors proved that to test the basicness of S it is sufficient to test the basicness of the intersections of S with every irreducible surface. In fact, this is the best possible result about the dimension, but it was conjectured by the authors that the first obstruction to s-basicness should be recognized in dimension s+1. In the present paper, the authors confirm this conjecture, and in fact they prove the following result: if S is basic, then S is s-basic if and only if for every irreducible subset Y ofX of dimension s+1 the intersection S \Y is generically s-basic. The proof of this theorem requires a deep analysis of real valuations. The results of independent interest about fans (algebroid fans) defined by means of power series and the approximation problem solved in this case are the main tools of the proof.

The Number of Defining Equations of Affine Algebraic Sets

The Number of Defining Equations of Affine Algebraic Sets