Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry

Volume 41 (2000)

Abstracts

W. Benz: Eine gemeinsame Kennzeichnung der Krümmungsachse bei Regelflächen und Kurven, 41 (2000) 001--006

In einer Vorlesung über Differentielle Liniengeometrie im WS 1998/99, einem Gebiet, das so bedeutende Mathematiker wie William Rowan Hamilton (1805--1865), Eduard Kummer (1810--1893), Julius Plücker (1801--1868), Alfred Clebsch (1833--1872), Felix Klein (1849--1925), Eduard Study (1862--1930) schon im 19. Jahrhundert zur hohen Blüte geführt haben, erschienen mir immer wieder Rückgriffe auf Ansätze und Motivationen dieser frühen Zeit als höchst aufschlussreich, und dann schliesslich als wohl eigentlich unverzichtbar zum Verständnis dieser grossartigen geometrischen Disziplin. Im Zuge eines solchermassen beeinflussten Aufbaus meiner Vorlesung suchte ich eine einfache und natürliche Charakterisierung der Krümmungsachse bei Regelflächen, und dies in der Nähe einer ebensolchen bei Kurven. Das Resultat, das ich weder für Regelflächen noch für Kurven in der Literatur fand, ist in dieser Arbeit wiedergegeben.

A. Blunck: Reguli and Chains over Skew Fields, 41 (2000) 007--022

We give a geometric definition of a regulus  in a not necessarily pappian projective space of arbitrary dimension, such that the geometry of all reguli is a certain  chain geometry.

R. C. Laubenbacher, C. J. Woodburn: A New Algorithm for the Quillen-Suslin Theorem, 41 (2000) 023--032

We present a new algorithm for the Quillen-Suslin Theorem for polynomial rings.

M. R. Darafsheh, A. Rahnamai Barghi: Towards a Classification of Homogeneous Integral Table Algebras of Degree Five, 41 (2000) 033--048

We classify some of the homogeneous integral table algebras of degree 5 containing a faithful element of width 3 such that in a given basis of the algebra the basis elements are standard. This work is towards classifing homogeneous integral table algebras of degree 5.

M. Henk, R. Weismantel: On Minimal Solutions of Diophantine Equations, 41 (2000) 049--056

This paper investigates the region in which all the minimal solutions of a linear diophantine equation lie. We present best possible inequalities which must be satisfied by these solutions and thereby improve earlier results.

R. Steinwandt, J. Müller-Quade: Freeness, Linear Disjointness and Implicitization -- a Classical Approach, 41 (2000) 057--066

A method for deciding linear disjointness and freeness of finitely generated extension fields is given. The techniques used are based on a classical description of linear disjointness and on the Chow form. The required Gröbner basis techniques do not depend on tag variables. Finally, the solution obtained can also be used to solve the implicitization problem without involving tag variables.

S. Geller, A. K. Maloo, D. P. Patil, L. G. Roberts: Kähler Differentials of Affine Monomial Curves, 41 (2000) 067--084

We study the module of Kähler differentials of the co-ordinate ring A of an affine monomial curve over a field k of characteristic 0 (that is, A is the subalgebra of k[t] generated over k by various powers of t). We concentrate on the case where the exponents of the generators are in arithmetic progression, where we obtain a complete description of the Hilbert function H(n) of the differentials of A in terms of certain partitions of the integer n.

B. M. Vernikov: Modular Elements in Congruence Lattices of G-sets, 41 (2000) 085--092

A unary algebra is called a G-set if the set of unary operations is equipped by a group structure compatible with the unary structure. Modular elements in congruence lattices of G-sets are described.

F. Juhnke, O. Sarges: Minimal Spherical Shells and Linear Semi-Infinite Optimization, 41 (2000) 093--106

We prove some characterizing properties of the minimal shell of a convex body by means of linear semi-infinite optimization. Further we present a representation of the optimal solution of the corresponding optimization problem in dependence of the values of the support function of certain points of contact of the convex body and its minimal shell.

H. Groemer: On the Symmetric Difference Metric for Convex Bodies, 41 (2000) 107--114

Explicit inequalities are proved that relate the Hausdorff metric for convex bodies with the symmetric difference metric and a corresponding surface area deviation measure.

T. Shoji: Green Functions and a Conjecture of Geck and Malle, 41 (2000) 115--140

For any special unipotent class C of a split reductive group G over a finite field F_q, a special piece C* is defined. It is known that the cardinality of C*(F_q) is a polynomial in q. Geck and Malle proposed a conjectural algorithm for computing these polynomials, and verified it in the case of exceptional groups. In this paper we show, in the case of classical groups, that their conjecture is reduced to another conjecture concerning Springer representations of W. We verify this conjecture in the case where G is of type B_n,C_n or D_n with n<=6.

A. O. Kuku: Equivariant Higher K-Theory for Compact Lie Group Actions, 41 (2000) 141--150

The aim of this paper is to construct an equivariant higher K-theory for compact Lie group actions in a way analogous to the ones constructed in [A], [B] for finite groups and [C] for profinite groups.
[A] Dress, A.; Kuku, A. O.: A convenient setting for equivariant higher algebraic K-theory, Lecture Notes 966, Springer Verlag 1982, 59--68.
[B] Dress, A.; Kuku, A. O.: The Cartan map for equivariant higher algebraic K-groups, Communications in Algebra 9 (7) (1981) 727--746.
[C] Kuku, A. O.: Equivariant K-theory and the cohomology of profinite groups, Lecture Notes 1046, Springer Verlag 1984, 234--244.

S. Mehlhos: Simple Counter Examples for the Unsolvability of the Fermat- and Steiner-Weber-Problem by Compass and Ruler, 41 (2000) 151--158

The purpose of this short note is to give counter examples for the unsolvability of the Fermat- and Steiner-Weber-problem by compass and ruler. The used point sets made it possible to obtain for the Fermat - problem polynomials of the degree 3 and 4. Thus, for these counter examples Galois theory and computer algebra is not necessary. In the second part is given a counter example for the construction of the true length of Steiner trees in the three-dimensional space.

R. Walter: Homogeneous Parabolic Surfaces in R⁴, 41 (2000) 159--180

Parabolic surfaces in R⁴ bear their name because they satisfy parabolic partial differential equations. In this paper we shall determine and classify all parabolic non-ruled surfaces in R⁴ which are homogeneous in the sense of equiaffine geometry. In the course of the general discussion it will also come out that there are no compact parabolic surfaces, thus answering a question which has been open for some time.

G. Heinz: The Concept of Cutting Vectors for Vector Systems in the Euclidean Plane, 41 (2000) 181--194

The concept of cutting vectors is introduced to study the combinatorial structure of vector configurations in the plane. We give a full characterization of those n-tuples of integers which are realizable as cutting vector. Furthermore, "simple" mappings are defined that transform cutting vectors into each other. Thus it becomes possible to consider all combinatorial types of vector systems.

A. Thoma: Construction of Set Theoretic Complete Intersections via Semigroup Gluing, 41 (2000) 195--198

We present a very simple, but powerful, technique for finding monomial varieties which are set theoretic complete intersections. The technique is based on the concept of gluing semigroups that was defined by J. C. Rosales and used by K. Fischer, W. Morris and J. Shapiro to characterize complete intersection affine semigroup rings. There are several techniques in the literature proving that certain varieties are set theoretic complete intersections but all of them preserve the dimension of the variety and are mainly results about curves. The technique presented here does not preserve necessarily the dimension of the variety and it can combine the known results to produce set theoretic complete intersection varieties of any dimension, see Examples 4 and 5.

V. Pambuccian: Affine Super-Pythagorean Geometry, 41 (2000) 199--202

Super-Pythagorean fields can be thought of as arising from a weakening of the requirement of Euclideanity, while preserving the requirement of Pythagoreanity. Given that Pythagorean and Euclidean fields arise naturally as coordinate fields of planes with free mobility, and of those satisfying the circle axiom, it is natural to ask whether super-Pythagorean fields have a natural geometric meaning.
In [P] we have provided axiom systems for Euclidean planes with super-Pythagorean coordinate fields with or without order. Two axioms of those axiom systems, A7 and A8, are artificial, in the  sense that they do not reflect simple geometric intentions. This shortcoming will be remedied in this paper, where we shall display two axioms with clear geometric content to replace the above two. Moreover, in their new formulations these axioms will be affine statements.
[P] Pambuccian, V.: Euclidean superpythagorean geometry, Beitr. Algebra Geom. 39 (2), (1998) 255--258.

S. Benayadi: The Root Space Decomposition of the Quadratic Lie Superalgebras, 41 (2000) 203--222

A quadratic Lie superalgebra is a Lie superalgebra {\f g}={\f g}_{\bar 0}\oplus{\f g}_{\bar 1} with a non-degenerate, supersymmetric, even and {\f g}-invariant bilinear form B, B is called an invariant scalar product of {\f g}. In this paper, we study properties of the  decomposition of a quadratic Lie superalgebra {\f g}={\f g}_{\bar 0} \oplus{\f g}_{\bar 1} relative to a fixed Cartan subalgebra of {\f g}_{\bar 0}. Finally, we give two characterizations of the basic classical Lie superalgebras among the quadratic Lie superalgebras.

J. Brinkman: Generators of Order Two for S_n and its Two Double Covers, 41 (2000) 223--232

This paper considers the minimum number of involutions, i(G), required to generate both of the double covers G of the symmetric group. In particular, explicit generators, of order two, for each of the groups are also introduced. These generators may, for example, be useful for implementation in Magma-Cayley or GAP.

R. Sulanke: Möbius Invariants for Pairs of Spheres (S₁^m, S₂^l) in the Möbius Space Sⁿ, 41 (2000) 233--246

We construct a complete system of Möbius-geometric invariants for pairs (S^m, S^l), l <= m, of spheres contained in the Möbius space Sⁿ. It consists of n-m generalized stationary angles. We interpret these invariants geometrically.

K. Metsch, L. Storme: 2-Blocking Sets in PG(4,q), q Square, 41 (2000) 247--256

We show that the three smallest minimal point sets of PG(4,q), q square, q>9, that meet all planes are the set of points of a plane, the set of points in a Baer cone and the set of points in a Baer subgeometry PG(4,\sqrt q). This implies that PG(4,\sqrt q) is the unique smallest example of a set of points of PG(4,q) that meets every plane and contains no line. It also implies that PG(4,\sqrt q) is the unique smallest minimal set of points of PG(4,q) that meets all planes and generates PG(4,q).

A. G. Naoum, L. S. M. Al-Shalgy: Quasi-Frobenius Modules, 41 (2000) 257--266

Let R be a commutative ring with 1, and let M be a faithful R-module. We say that M is a quasi-Frobenius (in short QF) module if Hom_R(P,M) is either zero or a simple R-module for each simple R-module P. We give some characterizations of QF modules and we study the relation between QF modules and multiplication modules.

A. Boelcskei: Classification of Unilateral and Equitransitive Tilings by Squares of Three Sizes, 41 (2000) 267--278

After a break of 20 years and with the help of the fundamental book [A], the study of unilateral, equitransitive tilings of the plane by squares of three sizes was revived. First D. Schattschneider had found five possible tilings [A, p. 76] and Martini, Makai and Soltan generally characterized the unilateral tilings and obtained a new equitransitive arrangement [B]. They also described two other tilings constructed by B. Grünbaum. The problem to describe all possibilities remained open.
We shall derive all the unilateral and equitransitive tilings using the classification of the fundamental planigons. We prove, that only the eight known tilings are possible. We have learned that D. Schattschneider parallelly solved this problem, too. Our method is different from hers.
[A] Grünbaum, B.; Shepard, G. C.: Tilings and Patterns, W. H. Freeman 1987, 72--81.
[B] Martini, H.; Makai, E.; Soltan, V.: Unilateral Tilings of the Plane with Squares of Three Sizes, Beitr. Algebra Geom. 39 (1998) 481--495.

A. Alzati, G. M. Besana: Koszul Cohomology and k-Normality of a Projective Variety, 41 (2000) 279--290

Let X be a smooth complex projective variety and let L be a very ample line bundle on X, giving an embedding into N-dimensional complex projective space.Such an X is said to be k-normal, in the given embedding, if hypersurfaces of degree k cut on X a complete linear system. The Koszul groups of X are utilized to obtain information about its k-normality. Results in this direction are reached via establishing an upper bound for the the degree of the generators of the ring of global sections of all tensor powers of L. The above idea, paired with existing results due to Green and Butler, is applied to special classes of varieties, including scrolls over curves and surfaces.

T. Holm: Hochschild Cohomology Rings of Algebras k[X]/(f), 41 (2000) 291--302

Let k be a commutative ring with unity and let f be a monic polynomial from k[X]. We determine the ring structure of the Hochschild cohomology for the k-algebra k[X]/(f). This generalizes results of [A] on the Hochschild cohomology rings of modular group algebras of cyclic groups over fields.
[A] Cibils, C.; Solotar, A.: Hochschild cohomology algebra and Hopf bimodules of an abelian group, Arch. Math. 68 (1997) 17--21.

G. Jones, M. Klin, F. Lazebnik: Automorphic Subsets of the n-Dimensional Cube, 41 (2000) 303--323

An automorphic subset of the n-dimensional cube Q_n is an orbit of a subgroup of Aut(Q_n), acting on the vertices. We develop a theory of such subsets, and we show that those containing 0 coincide with the cwatsets introduced by Sherman and Wattenberg in response to a statistical result of Hartigan. Using this characterisation, together with results from finite group theory and number theory, we answer two questions on cwatsets posed by Sherman and Wattenberg, and we complete the proofs of some results outlined by Kerr.

A. V. Kelarev: Varieties and Sums of Rings, 41 (2000) 325--328

For commutative algebras over a field, we describe all varieties closed for taking algebras which are sums of their two subalgebras in the variety.

A. Schürmann: On Parametric Density of Finite Circle Packings, 41 (2000) 329--334

The parametric density of a finite circle packing X_n + D is defined as ratio of areas n times A(D) / A(conv (X_n + r D)). We show that densest finite lattice packings are attained in a critical lattice. Further, we give an upper bound for the density with parameters r <= 3.232... which is attained by Groemer packings. Moreover, we show that the given bound holds for arbitrary finite circle packings and parameters less than 1 as a consequence of known results by H. Groemer ["Uber die Einlagerung von Kreisen in einen konvexen Bereich", Math. Z. 73 (1960) 285--294] and G. Wegner ["Uber endliche Kreispackungen in der Ebene, Studia Sci. Math. Hung. 21 (1986) 1--28].

M. Meyer, S. Reisner: A Geometric Proof of Some Inequalities Involving Mixed Volumes, 41 (2000) 335--344

A new, "spherical harmonics free" proof of mixed-volume inequalities due to Schneider and to Goodey and Groemer, is presented.

J. D. Velez, R. Florez: Failure of Splitting from Module-Finite Extension Rings, 41 (2000) 345--357

 

S. Carter, Y. Kaya: Immersions with a Parallel Normal Field, 41 (2000) 359--370

We investigate the restrictions which are placed on the focal set of a submanifold of Euclidean space under the assumption that the submanifold has a parallel normal field.

A. J. Breda d'Azevedo, G. A. Jones: Double Coverings and Reflexive Abelian Hypermaps, 41 (2000) 371--389

We describe a general theory of hypermaps on surfaces, possibly non-orientable or with boundary. This includes techniques for constructing hypermaps as products and as double coverings, and for representing hypermaps as maps using homomorphisms between extended triangle groups. As a corollary we obtain a classification of the 16 reflexible hypermaps with abelian automorphism groups.

A. Kemnitz, L. Szabo, Z. Ujvary-Menyhart: Protecting Regular Polygons, 41 (2000) 391--399

The minimum number of mutually non-overlapping congruent copies of a convex body K so that they can touch K and prevent any other congruent copy of K from touching K without overlapping each other is called the protecting number of K. We prove that the protecting number of any regular polygon is three or four, and both values are indeed attained.

F. Fodor: The Densest Packing of 12 Congruent Circles in a Circle, 41 (2000) 401--409

The densest packings of n congruent circles in a circle are known for n <= 11 and n = 19. We exhibit here the densest packing of 12 congruent circles in a circle. In fact, we show that the optimal configuration is the same as the one S. Kravitz ["Packing cylinders into cylindrical containers, Math. Mag. 40 (1967) 65--71] conjectured. We use a technique developed from a method of P. Bateman and P. Erdös ["Geometrical extrema suggested by a lemma of Besicovitch", Amer. Math. Monthly 58 (1951) 306--314].

K. Bezdek: On Rhombic Dodecahedra, 41 (2000) 411--416

We prove that the intrinsic i-volume of any d-dimensional zonotope generated by d + 1 (resp. d) line segments and containing a d-dimensional unit ball in E^d is at least as large as the intrinsic i-volume of the d-dimensional regular zonotope generated by d + 1 line segments having inradius 1, where i = 1, ... , d - 1, d.

B. Weissbach: Sets with Large Borsuk Number, 41 (2000) 417--423

We construct sets in Euclidean spaces of dimension d given by the binomial coefficient (4m-2) over 2, where m is a power of a prime, with the property that they can only be covered with a large number of sets having smaller diameter. Thereby we generalize a result of A. M. Raigorodskii and, in addition, we prove that there exists a counterexample to the so called "Borsuk-conjecture" already in dimension (34 over 2) - 1 = 560.

B. Weissbach: On a Covering Problem in the Plane, 41 (2000) 425--426

We make a remark to a covering problem stated by V. Boltyanski, H. Martini and P. S. Soltan ["Excursions into combinatorial geometry", Springer-Verlag, 1997].

M. Schaaf: Decomposition of Isometric Immersions between Warped Products, 41 (2000) 427--436

A decomposition theorem for isometric immersions between two arbitrary warped products is derived. Thereby the well known decomposition theorems of Moore and Nölker are generalized.

U. Brehm, J. Voigt: Asymptotics of Cross Sections for Convex Bodies, 41 (2000) 437--454

For normed isotropic convex bodies in Rⁿ we investigate the behaviour of the (n-1)-dimensional volume of intersections with hyperplanes orthogonal to a fixed direction, considered as a function of the distance of the hyperplane to the origin. It is a conjecture that for arbitrary normed isotropic convex bodies and random directions this function -- with high probability -- is close to a Gaussian density, for large dimension n. This would be a kind of central limit theorem. We determine this function explicitly for several families of convex bodies and several directions and obtain results concerning the asymptotic behaviour supporting the conjecture.

S. Blum: Initially Koszul Algebras, 41 (2000) 455--467

Let R = K[X₁, ... , X_n]/I be a homogeneous K-algebra where K is a field and I an ideal generated by quadrics. Conca, Trung and Valla have introduced the concept of Koszul filtrations to verify the Koszul property for certain algebras. We call a K-algebra R "initially Koszul" if R admits a Koszul filtration consisting of a flag F = {(x₁, ... , x_i) | i = 0, ... , n} where x₁, ... , x_n is a sequences of 1-forms in R. For such algebras the defining ideal I has a quadratic Groebner basis which has been proved by Conca, Rossi and Valla.
We show that initially Koszulness is equivalent to a certain exchange property of the corresponding initial ideal of I and that this property is preserved under some ring operations. Generic flags form Koszul filtrations when I has a 2-linear resolution over the polynomial ring. Provided K is algebraically closed and char(K) is not equal to 2 we classify all algebras which are initially Koszul with respect to any K-basis of R₁. Initially Koszul semigroup rings R have shellable divisor posets.

J. Quistorff: On Codes with Given Minimum Distance and Covering Radius, 41 (2000) 469--478

Codes with minimum distance at least d and covering radius at most d - 1 are considered. The minimal cardinality of such codes is investigated. Herewith, their connection to covering problems is applied and a new construction theorem is given. Additionally, a new lower bound for the covering problem is proved. A necessary condition on an existence problem is presented by using a multiple covering of the farthest-off points.

R. Riesinger: The Second Extension of the Thas-Walker Construction, 41 (2000) 479--488

We extend the extended Thas-Walker construction by introducing the concept "flocklet" of the Klein quadric. Thus we have a tool to construct spreads with asymplecticly complemented regulization.

S. Veldsman: Polynomial and Transformation Composition Rings, 41 (2000) 489--511

The base of an arbitrary composition ring is defined. This is used, amongst others, to identify polynomial type composition rings and to describe the maximal ideals of certain types of composition rings.

C. U. Sanchez, A. N. Garcia, W. Dal Lago: Planar Normal Sections on the Natural Embedding of a Real Flag Manifold, 41 (2000) 513--530

This paper contains a proof of the following result which is an extension of the main result in a previous paper of the authors ["Planar Normal Sections on the Natural Imbedding of a Flag Manifold", Geom. Dedicata 53 (1994) 223--235], to the general case of real flag manifolds (also called R-spaces or orbits of s-representations).

THEOREM. Let M be a real flag manifold and let j: M --> p its canonical embedding. Let the subset X[M] of RP^n-1, (n = dim M), be the variety of directions of pointwise planar normal sections at a point of M and let the subset X_c[M] of CP^n-1 be the natural complexification of X[M]. Let c denote the Euler-Poincare characteristic. Then:
(i) c(X[M]) = c(RP^n-1) = 0, if n is even, and = 1, if n is odd.
(ii) c(X_c[M]) = c(CP^n-1) = n.

L. F. Matusevich: Rational Summation of Rational Functions, 41 (2000) 531--536

We characterize rational functions for which their indefinite sum is again a rational function.

L. Haddad, D. Lau: Pairwise Intersections of Slupecki Type Maximal Partial Clones, 41 (2000) 537--555

Let k >= 2 and K be a k-element set. We study the pairwise intersections of all maximal partial clones of Slupecki type on K. More precisely, we show that with one exception, if M and M' are two strong maximal partial clones of Slupecki type on K, then the intersection of M and M' is covered by both M and M'. We also show that the situation is quite different if the non-strong maximal partial clone on K is involved in the intersection.

H. Salzmann: On the Classification of 16-Dimensional Planes, 41 (2000) 557--568

A topological projective plane with a compact point space P of finite positive covering dimension has lines of a dimension dividing 8, and the dimension of P is twice the dimension of a line. The classification problem requires to determine all such planes having a sufficiently large group of automorphisms. The cases where the dimension of P is at most 8 are understood fairly well. Here we prove the following result for 16-dimensional planes: "A connected automorphism group of dimension at least 33 is semi-simple or has a minimal normal vector subgroup consisting of axial collineations, or the plane is a Hughes plane". Based on this theorem, a complete classification has been obtained of all planes admitting a group of dimension at least 35 fixing exactly one line and no point; see in particular H.  Hähl, Geom. Dedic. 83 (2000) 105--117.

M. Petrich, P. V. Silva: Relatively Free *-Bands, 41 (2000) 569--588

A *-band is an algebra consisting of a band (idempotent semigroup) on which an involution * is defined satisfying an extra condition; in summary

(xy)* = y*x*,   x** = x,   x = xx*x,   (xy)z = x(yz),   x² = x.

The lattice of all *-band varieties was determined by Adair who also provided a basis for the identities of each variety. Another system of bases was devised by Petrich. Defining certain operators on the free involutorial semigroup F on a nonempty set X, we construct a system of fully invariant congruences on F which is in bijection with the set of all proper *-band varieties, with the exception of normal *-band varieties which require a different treatment. The proof of this result is based on those evoked above and is broken into a long sequence of lemmas.

E. Hertel: Self-Similar Simplices, 41 (2000) 589--595

A Euclidean d-simplex S is called k-self-similar if S can be dissected into k simplices (k >= 2), each similar to S. Each triangle (d=2) is k-self-similar for k=4 and k >= 6 whereas for d>2 most d-simplices are not self-similar. A first class of 3-simplices which are m³-self-similar for all positive integers m is characterized.