Algebraic points on the plane

This article gives quantitative estimates for the measure of points (x, y) of a given rectangle admitting the construction of polynomials P(t) with small (with respect to the height of the polynomial) values of P(x) and P(y). Such estimates can be used in the problem of distribution of algebraic points on the plane. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 73–80, 2005.     

Algebraic structures on graphs

In this paper we consider directed graphs with algebraic structures: group-graphs, ringgraphs, involutorial graphs, affine graphs, graphs of morphisms between graphs, graphs of reduced paths of an involutorial graph, etc. We show also how several well-known algebraic constructions can be carried over to graphs. As a typical example we generalize the construction of the group of automorphisms of a set, by constructing a group-graph associated with any given graphΓ. It is the group-graph of reduced paths of the involutorial graph associated to the graph of automorphisms ofΓ.     

Algebraic properties of classes of path ideals of posets

We consider path ideals associated to special classes of posets such as tree posets and cycles. We express their property of being sequentially Cohen–Macaulay in terms of the underlying poset. Moreover, monomial ideals, which arise in algebraic statistics from the Luce-decomposable model and the ascending model, can be viewed as path ideals of certain posets. We study invariants of these so-called Luce-decomposable monomial ideals and ascending ideals for diamond posets and products of chains. In particular, for these classes of posets, we explicitly compute their Krull dimension, their projective dimension, their Castelnuovo–Mumford regularity and their Betti numbers.     

Algebraic solutions of plane vector fields

We present an algorithm that can be used to check whether a given derivation of the complex affine plane has an invariant algebraic curve and discuss the performance of its implementation in the computer algebra system Singular.     

A self paired Hopf algebra on double posets and a Littlewood-Richardson rule

Let D be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On ZD we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations ZD is a Hopf algebra. We define a symmetric bilinear form on this Hopf algebra: it counts the number of pictures (in the sense of Zelevinsky) between two double posets. This form is a Hopf pairing, which means that product and coproduct are adjoint each to another. The product and coproduct correspond respectively to disjoint union of posets and to a natural decomposition of a poset into order ideals. Restricting to special double posets (meaning that the second order is total), we obtain a notion equivalent to Stanley's labelled posets, and a Hopf subalgebra already considered by Blessenohl and Schocker. The mapping which maps each double poset onto the sum of the linear extensions of its first order, identified via its second (total) order with permutations, is a Hopf algebra homomorphism, which is isometric and preserves the internal product, onto the Hopf algebra of permutations, previously considered by the two authors. Finally, the scalar product between any special double poset and double posets naturally associated to integer partitions is described by an extension of the Littlewood-Richardson rule.     

Webs and projective structures on a plane

We prove that there is a one-to-one correspondence between projective structures defined by torsion-free connections with skew-symmetric Ricci tensor and Veronese webs on a plane. The correspondence is used to characterise the projective structures in terms of second order ODEs.     

Algebraic isomorphisms and strongly double triangle subspace lattices

Let D={{0},K,L,M,X} be a strongly double triangle subspace lattice on a non-zero complex reflexive Banach space X, which means that at least one of three sums K + L, L + M and M + K is closed. It is proved that a non-zero element S of AlgD is single in the sense that for any A,B∈AlgD, either AS = 0 or SB = 0 whenever ASB = 0, if and only if S is of rank two. We also show that every algebraic isomorphism between two strongly double triangle subspace lattice algebras is quasi-spatial.     

Curves in the double plane

We study in detail locally Cohen-Macaulay curves in P⁴ which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes H _d,g(2H) of lo-cally Cohen-Macaulay curves in 2H of degree d and arithmetic genus g, and we show that H _d,g(2H) is connected. We also discuss the Rao module of these curves and liaison and biliaison equiva-lence classes.     

Harmonics on posets

Given a finite ranked poset P, for each rank of P a space of complex valued functions on P called harmonics is defined. If the automorphism group G of P is sufficiently rich, these harmonic spaces yield irreducible representations of G. A decomposition theorem, which is analogous to the decomposition theorem for spherical harmonics, is stated. It is also shown that P can always be decomposed into posets whose principal harmonics are orthogonal polynomials. Classical examples are given.     

III-Homomorphisms and III-congruences on posets

A new homomorphism between two partially ordered sets (the III-homomorphism) and a new congruence on a poset (the III-congruence) are introduced. Some properties of these homomorphisms and congruences and their relationship to the other known homomorphisms and congruences on posets are investigated. In contrast to total algebras, there are many different ways to introduce these notions. It is usually required that the respective notions should coincide with the usual definitions whenever lattices or semilattices are treated. The present paper presents an approach which in some sense completes the hierarchy of definitions so far used.     

Characterizations of the convex geometries arising from the double shellings of posets

We investigate the class of double-shelling convex geometries. A double-shelling convex geometry is the collection of sets represented as the intersection of an ideal and a filter of a poset. The size of the stem of any rooted circuit of a double-shelling convex geometry is 2. We characterize the double-shelling convex geometries by the conditions that the rooted circuits should fulfill. Moreover we also characterize the same class in terms of trace-minimal forbidden minors.     

Algebraic and geometric structures of analytic partial differential equations

We study the problem of the compatibility of nonlinear partial differential equations. We introduce the algebra of convergent power series, the module of derivations of this algebra, and the module of Pfaffian forms. Systems of differential equations are given by power series in the space of infinite jets. We develop a technique for studying the compatibility of differential systems analogous to the Gröbner bases. Using certain assumptions, we prove that compatible systems generate infinite manifolds.     

Algebraic structures on Grothendieck groups of a tower of algebras

The Grothendieck group of the tower of symmetric group algebras has a self-dual graded Hopf algebra structure. Inspired by this, we introduce by way of axioms, a general notion of a tower of algebras and study two Grothendieck groups on this tower linked by a natural paring. Using representation theory, we show that our axioms give a structure of graded Hopf algebras on each Grothendieck groups and these structures are dual to each other. We give some examples to indicate why these axioms are necessary. We also give auxiliary results that are helpful to verify the axioms. We conclude with some remarks on generalized towers of algebras leading to a structure of generalized bialgebras (in the sense of Loday) on their Grothendieck groups.     

Some theorems on graphs and posets

In this journal, Leclerc proved that the dimension of the partially ordered set consisting of all subtrees of a tree T, ordered by inclusion, is the number of end points of T. Leclerc posed the problem of determining the dimension of the partially ordered set P consisting of all induced connected subgraphs of a connected graph G for which P is a lattice.In this paper, we prove that the poset P consisting of all induced connected subgraphs of a nontrivial connected graph G, partially ordered by inclusion, has dimension n where n is the number of noncut vertices in G whether or not P is a lattice. We also determine the dimension of the distributive lattice of all subgraphs of a graph.     

Algebraic characterization of generic strongly semi-regular rational PH plane curves

In this paper, we introduce a new algebraic method to characterize rational PH plane curves. And using this method, we study the algebraic characterization of generic strongly regular rational plane PH curves expressed in the complex formalism which is introduced by R. T. Farouki. We prove that generic strongly semi-regular rational PH plane curves are completely characterized by solving a simple functional equation ${\mathcal{H}}(f,g) = h^2 $ whereh is a complex polynomial and ${\mathcal{H}}$ is a bi-linear operator defined by ${\mathcal{H}}(f,g) = f'g - fg'$ for complex polynomialsf, g.     

Cremona transformations and special double structures

We consider Cremona Transformations on , whose base locus schemes are double Fossum-Ferrand structures supported on a smooth, irreducible positive dimensional subvariety. We show that if the codimension of the base locus is 2 or if its dimension is no greater than , then N=3 and such a transformation is a Cubo-Cubic Cremona Transformation not defined along a twisted cubic curve. We also prove that the same conclusion holds for such Cremona Transformations either assuming Hartshorne Conjecture on Complete Intersections or that they are defined by degree three homogeneous polynomials.First author partially supported by DMAT/UFPE and by CNPq/Bolsa de Produtividade em Pesquisa.Second author partially supported by CNPQ/Bolsa de Produtividade em Pesquisa, and by PRONEX-FAPERJ/Geometria Algebrica e Algebra Comutativa.