A B S T R A C T S

FIBRÉE VECTOLIELS SEMI-STABLES SUR UNE COURBE DE GENDRE DEUX ET ASSOCIATION DES POINTS DANS L'ESPACE PROJECTIF
Cristian Anghel Cristian.Anghel@imar.ro

2000 Mathematics Subject Classification: 14D20, 14J60.
Key words: courbe projective, fibres semi-stables, association des points.

The aim of this paper is to prove that a certain involution on the moduli space of stable bundles on a curve of genus two, can be viewed as a geometric operation on the corresponding set of points in projective space.

Z₂-GRADED POLYNOMIAL IDENTITIES FOR SUPERALGEBRAS OF BLOCK-TRIANGULAR MATRICES
Onofrio M. Di Vincenzo divincenzo@dm.uniba.it

2000 Mathematics Subject Classification: Primary 16R50, Secondary 16W55.
Key words: Graded cocharacter sequence, triangular matrices, superalgebra, polynomial identity.

We present some results about the Z₂-graded polynomial identities of block-triangular matrix superalgebras

R = é ê ë A M 0 B ù ú û .

In particular, we describe conditions for the T₂-ideal of a such superalgebra to be factorable as the product T₂(A)T₂(B). Moreover, we give formulas for computing the sequence of the graded cocharacters of R in some interesting case.

AUTOMORPHISMS OF THE PLANAR TREE POWER SERIES ALGEBRA AND THE NON-ASSOCIATIVE LOGARITHM
L. Gerritzen lothar.gerritzen@ruhr-uni-bochum.de

2000 Mathematics Subject Classification: 17A50, 05C05.
Key words: planar rooted tree, planar formal power series, non-associative exponential series, substitution endomorphisms and automorphisms for planar power series, Hopf algebra.

In this note we present the formula for the coefficients of the substitution series f(g(x)) of planar tree power series g(x) into f(x). The coefficient c_T(f(g(x))) relative to a finite planar reduced rooted tree T is the sum

å cS(f) c(T-In(S)) (g)

which is extended over all open subtrees S of T. In this expression c_S(f) is the coefficient of f with respect to S and T-In(S) is the complement of the inner vertices of S in T while c_(T-In(S))(g) is the product of the coefficients of g relative to the connected components of the forest T-In(S).


Also we give the formula for the compositional inverse of a planar tree power series g in terms of coefficients of g. It is shown that the generic non-associative exponential series EXP has an inverse LOG. Results about coefficients of LOG are presented.


The group G = Aut(K {{x}}_¥) of automorphisms of the algebra K{{x}}_¥ of planar tree power series in a single variable x over a field K, see [6], is an affine group. The finite planar reduced rooted trees T are indexing a set of variables y_T of K[G] and the composition in G induces a coproduct

D: K[G] ® K[G] ÄK[G]

for which

D(yT) = å yS Äy(T-In(S))

where the summation is as in the formula above.


It is an interesting question how (K[G], D) which is indeed a Hopf algebra is related to the Hopf algebra of Kreimer, see [12], in which the coproduct on a variable y_T is defined by constructing admissible cuts on the set of edges of the planar rooted tree T while the above D(y_T) can be described by admissible cuts on the set of vertices of T.

ON THE FACTORIZATION OF THE POINCARÉ POLYNOMIAL: A SURVEY
Ersan Akyildiz ersan@metu.edu.tr

2000 Mathematics Subject Classification: 13P05, 14M15, 14M17, 14L30.
Key words: Factorization, Poincaré polynomial, Algebraic homogeneous spaces.

Factorization is an important and very difficult problem in mathematics. Finding prime factors of a given positive integer n, or finding the roots of the polynomials in the complex plane are some of the important problems not only in algorithmic mathematics but also in cryptography. For a given smooth m-dimensional real manifold X, one has the associated Poincaré polynomial P (X,t) = å_{i = 0}^m b_i (X) tⁱ of X, where b_i (X) = dim_R Hⁱ (X ; R) is the i-th Betti number of X. It is clear that the factorization of P (X,t) as series over the complex numbers C will carry lots of information about the topological and geometric invariants of X. This is possibly why a factorization of even such a special polynomial P (X,t) is expected to be hard. However we can still search for algorithms to write P (X,t) as a product of some nontrivial power series. One notes that the factorizations

P (Pn, t1/2) = n å i = 0    ti = 1-tn+1 1-t ,

P (G Ln /B, t1/2) = n Õ i = 1  1-ti 1-t

are examples of such kind. Here Pⁿ is the n-dimensional complex projective space and GL_n / B is the complex full flag manifold associated to the upper triangular matrices B in the invertible complex matrices GL_n. The aim of this survey article is to give first a direct self-contained elementary algebraic treatment of the problem and then provide examples of nonsingular complex projective varieties X so that the C-algebra H^* (X; C) fits into this treatment. This will allow us to factorize P (X,t) as above for such a variety X. These varieties X will include all the homogeneous spaces G/P, their smooth Schubert subvarieties and more. It is also interesting to note that in this approach, one can read off smoothness of a Schubert variety from the factorization of its Poincaré polynomial, which is discussed in Section 2 and 3.

DICKSON POLYNOMIALS THAT ARE PERMUTATIONS
Mihai Cipu mihai.cipu@imar.ro

2000 Mathematics Subject Classification: 11T06, 13P10.
Key words: Dickson polynomial, Gröbner basis, permutation polynomial.

A theorem of S.D. Cohen gives a characterization for Dickson polynomials of the second kind that permutes the elements of a finite field of cardinality the square of the characteristic. Here, a different proof is presented for this result.

EQUIMULTIPLE LOCUS OF EMBEDDED ALGEBROID SURFACES AND BLOWING-UP IN CHARACTERISTIC ZERO
R. Piedra-Sánchez piedra@algebra.us.es
J. M. Tornero tornero@algebra.us.es

2000 Mathematics Subject Classification: 14B05, 32S25.
Key words: Resolution of surface singularities, blowing-up, equimultiple locust.

The smooth equimultiple locus of embedded algebroid surfaces appears naturally in many resolution processes, both classical and modern. In this paper we explore how it changes by blowing-up.

COMPLEX HYPERBOLIC SURFACES OF ABELIAN TYPE
R.-P. Holzapfel holzapfl@mathematik.hu-berlin.de

2000 Mathematics Subject Classification: 11G15, 11G18, 14H52, 14J25, 32L07.
Key words: algebraic curve, elliptic curve, algebraic surface, Shimura variety, arithmetic group, Picard modular group, Gauß numbers, congruence numbers, Kähler-Einstein metric, negative constant curvature, unit ball.

We call a complex (quasiprojective) surface of hyperbolic type, iff - after removing finitely many points and/or curves - the universal cover is the complex two-dimensional unit ball. We characterize abelian surfaces which have a birational transform of hyperbolic type by the existence of a reduced divisor with only elliptic curve components and maximal singularity rate (equal to 4). We discover a Picard modular surface of Gauß numbers of bielliptic type connected with the rational cuboid problem. This paper is also necessary to understand new constructions of Picard modular forms of 3-divisible weights by special abelian theta functions.

INVOLUTION MATRIX ALGEBRAS - IDENTITIES AND GROWTH
Tsetska Grigorova Rashkova tcetcka@ami.ru.acad.bg

2000 Mathematics Subject Classification: 16R50, 16R10.
Key words: involution, polynomial identities, symmetric and skew-symmetric variables, Bergman type polynomials, characters, Hilbert series, codimensions, growth

The paper is a survey on involutions (anti-automorphisms of order two) of different kinds. Starting with the first systematic investigations on involutions of central simple algebras due to Albert the author emphasizes on their basic properties, the conditions on their existence and their correspondence with structural characteristics of the algebras.
Focusing on matrix algebras a complete description of involutions of the first kind on M_n(F) is given. The full correspondence between an involution of any kind for an arbitrary central simple algebra A over a field F of characteristic 0 and an involution on M_n(A) specially defined is studied.
The research mainly in the last 40 years concerning the basic properties of involutions applied to identities for matrix algebras is reviewed starting with the works of Amitsur, Rowen and including the newest results on the topic. The cocharactes, codimensions and growth of algebras with involutions are considered as well.

A SMOOTH FOUR-DIMENSIONAL G-HILBERT SCHEME
Magda Sebestean sebes@math.jussieu.fr

2000 Mathematics Subject Classification: 14C05, 14L30, 14E15, 14J35.
Key words: quotient singularities, crepant resolutions, toric varieties, $G$-Hilbert scheme, $G$-graph.

When the cyclic group G of order 15 acts with some specific weights on affine four-dimensional space, the G-Hilbert scheme is a crepant resolution of the quotient A⁴/G. We give an explicit description of this resolution using G-graphs.

COHOMOLOGY OF THE G-HILBERT SCHEME FOR ¹/_r(1,1,r-1)
Oskar Kedzierski oskar@impan.gov.pl

2000 Mathematics Subject Classification: Primary 14E15; Secondary 14C05,14L30.
Key words: McKay correspondence; resolutions of terminal quotient singularities;$G$-Hilbert scheme.

In this note we attempt to generalize a few statements drawn from the 3-dimensional McKay correspondence to the case of a cyclic group not in SL(3,C). We construct a smooth, discrepant resolution of the cyclic, terminal quotient singularity of type ¹/_r (1,1,r-1), which turns out to be isomorphic to Nakamura's G-Hilbert scheme. Moreover we explicitly describe tautological bundles and use them to construct a dual basis to the integral cohomology on the resolution.

MINIMAL CODEWORDS IN LINEAR CODES
Yuri Borissov youri@moi.math.bas.bg
Nickolai Manev nlmanev@moi.math.bas.bg

2000 Mathematics Subject Classification: 94B05, 94B15.
Key words: minimal codewords; cyclic codes; binary Reed-Muller code.

Cyclic binary codes C of block length n = 2^m-1 and generator polynomial g(x) = m₁(x)m_2^s+1(x), (s,m) = 1, are considered. The cardinalities of the sets of minimal codewords of weights 10 and 11 in codes C and of weight 12 in their extended codes ^(C) are determined.
The weight distributions of minimal codewords in the binary Reed-Muller codes RM(3,6) and RM(3,7) are determined. The applied method enables codes with larger parameters to be attacked.

KNESER AND HEREDITARILY KNESER SUBGROUPS OF A PROFINITE GROUP
Serban A. Basarab Serban.Basarab@imar.ro

2000 Mathematics Subject Classification: 20E18, 12G05, 12F10, 12F99.
Key words: Profinite group, Cogalois group of a field extension, Cogalois theory, continuous 1-cocycle, Kneser group of cocycles, Cogalois group of cocycles, radical subgroup, hereditarily radical subgroup, Kneser subgroup, almost Kneser subgroup, hereditarily Kneser subgroup, spectral space, coherent map.

Given a profinite group G acting continuously on a discrete quasi-cyclic group A, certain classes of closed subgroups of G (radical, hereditarily radical, Kneser, almost Kneser, and hereditarily Kneser) having natural field theoretic interpretations are defined and investigated. One proves that the hereditarily Kneser subgroups of G form a closed subspace of the irreducible spectral space of all closed subgroups of G, and a hereditarily Kneser criterion for hereditarily radical subgroups is provided.

LINEARLY NORMAL CURVES IN Pⁿ
Ovidiu Pasarescu ovidiu.pasarescu@imar.ro

2000 Mathematics Subject Classification: 14H45, 14H50, 14J26.
Key words: linearly normal curves, rational surfaces.

We construct linearly normal curves covering a big range from Pⁿ, n ³ 6 (Theorems 1.7, 1.9). The problem of existence of such algebraic curves in P³ has been solved in [4], and extended to P⁴ and P⁵ in [10]. In both these papers is used the idea appearing in [4] and consisting in adding hyperplane sections to the curves constructed in [6] (for P³) and [15, 11] (for P⁴ and P⁵) on some special surfaces. In the present paper we apply the same idea to the curves lying on some rational surfaces from Pⁿ, constructed in [12, 3, 2] (see [13, 14] also).

HENSELIAN DISCRETE VALUED FIELDS ADMITTING ONE-DIMENSIONAL LOCAL CLASS FIELD THEORY
I. D. Chipchakov chipchak@math.bas.bg

2000 Mathematics Subject Classification: 11S31 12E15 12F10 12J20.
Key words: field admitting (one-dimensional) local class field theory, strictly primarily quasilocal field, Henselian valued field, Brauer group, character group, norm group, Galois extension, regular group formation.

This paper gives a characterization of Henselian discrete valued fields whose finite abelian extensions are uniquely determined by their norm groups and related essentially in the same way as in the classical local class field theory. It determines the structure of the Brauer groups and character groups of Henselian discrete valued strictly primary quasilocal (or PQL-) fields, and thereby, describes the forms of the local reciprocity law for such fields. It shows that, in contrast to the special cases of local fields or strictly PQL-fields algebraic over a given global field, the norm groups of finite separable extensions of the considered fields are not necessarily equal to norm groups of finite Galois extensions with Galois groups of easily accessible structure.

INVARIANTS OF UNIPOTENT TRANSFORMATIONS ACTING ON NOETHERIAN RELATIVELY FREE ALGEBRAS
Vesselin Drensky drensky@math.bas.bg

2000 Mathematics Subject Classification: 16R10, 16R30.
Key words: Noncommutative invariant theory; unipotent transformations; relatively free algebras.

The classical theorem of Weitzenböck states that the algebra of invariants K[X]^g of a single unipotent transformation g Î GL_m(K) acting on the polynomial algebra K[X] = K[x₁,¼,x_m] over a field K of characteristic 0 is finitely generated. This algebra coincides with the algebra of constants K[X]^d of a linear locally nilpotent derivation d of K[X]. Recently the author and C. K. Gupta have started the study of the algebra of invariants F_m(V)^g where F_m(V) is the relatively free algebra of rank m in a variety V of associative algebras. They have shown that F_m(V)^g is not finitely generated if V contains the algebra UT₂(K) of 2×2 upper triangular matrices (and g ¹ 1). The main result of the present paper is that the algebra F_m(V)^g is finitely generated if and only if the variety V does not contain the algebra UT₂(K). As a by-product of the proof we have established also the finite generation of the algebra of invariants T_nm^g where T_nm is the mixed trace algebra generated by m generic n×n matrices and the traces of their products.

REMARKS ON THE NAGATA CONJECTURE
Beata Strycharz-Szemberg bm0061@uni-essen.de
Tomasz Szemberg mat905@uni-essen.de

2000 Mathematics Subject Classification: 14C20, 14E25, 14J26.
Key words: Nagata Conjecture, linear series, Seshadri constants, Harbourne-Hirschowitz Conjecture, big divisors.

The famous Nagata Conjecture predicts the lowest degree of a plane curve passing with prescribed multiplicities through given points in general position. We explain how this conjecture extends naturally via multiple point Seshadri constants to ample line bundles on arbitrary surfaces. We show that if there exist curves of unpredictable low degree, then they must have equal multiplicities in all but possibly one of the given points. We use this restriction in order to obtain lower bounds on multiple point Seshadri constants on a surface. We discuss also briefly a seemingly new point of view on the Nagata Conjecture via the bigness of the involved linear series.

BINOMIAL SKEW POLYNOMIAL RINGS, ARTIN-SCHELTER REGULARITY, AND BINOMIAL SOLUTIONS OF THE YANG-BAXTER EQUATION
Tatiana Gateva-Ivanova tatianagateva@yahoo.com, tatyana@aubg.bg, tatiana@math.bas.bg

2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.
Key words: Yang-Baxter equation, Quadratic algebras, Artin-Schelter regular rings, Quantum groups.

Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A^! is Frobenius of dimension n, with a regular socle and for each x,y Î X an equality of the type xyy = azzt, where a Î k \{0}, and z,t Î X is satisfied in A. We prove the equivalence of the notions a binomial skew polynomial ring and a binomial solution of YBE. This implies that the Yang-Baxter algebra of such a solution is of Poincaré-Birkhoff-Witt type, and possesses a number of other nice properties such as being Koszul, Noetherian, and an Artin-Schelter regular domain.