Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 43 (2002)
Abstracts
B. Haas:
A Simple Counterexample to Kouchnirenko's Conjecture,
43 (2002) 001--008
- In the late 70's A. Kouchnirenko posed the problem of bounding from above the number
of positive real roots (that is, with positive coordinates) of a system of k polynomial
equations in k variables not in function of the degrees of the polynomials (like in Bezout
Theorem), but in function of the number of terms involved, and he conjectured an upper bound.
Although he never wrote this conjecture (as far as I know), many references to it can be found
[A. G. Khovanskii, Dokl. Akad. Nauk SSSR 255(4) (1980) 804--807] for one of the first and
[B. Sturmfels, Amer. Math. Monthly 105(10) (1998) 907--922] for one of the most recent).
I recently learned that Kouchnirenko himself was "100% sure" that his conjecture was false,
since a colleague of his once presented him a simple counterexample, a system of two threenomial
(polynomials with three terms) equations in two variables with 5 positive roots while the
conjecture predicts at most 4. His colleague tragically died soon afterward. The counterexample
was lost and Kouchnirenko never found it again. Here such a counterexample is presented, maybe
the lost counterexample found again ...
- For a Deligne-Lusztig variety X (w) arising from one of the classical (possibly twisted)
groups, we show that the Picard group of X(w) is generated by the finitely many Deligne-Lusztig
subvarieties of X (w). It is conjectured that this more generally should hold in any codimension,
and a good deal of supporting evidence for this claim is presented.
- Let G = (V, E, d) be any connected weighted graph which admits not only degenerated
realisations in the n-dimensional Euclidean space. Its configuration space is always
homeomorphic to a ((1/2)n(n+1) - e)-dimensional sphere, where n is the number of vertices
minus one and e the number of edges.
- [Abstract-pdf]
Using a result from Herzog [H] we prove the following. Let $(B_0,\hbox{\eufm n}_0)$ be
an artinian local algebra of embedding dimension $v$ over some field $L$ with tangent cone
$gr(B_0)\cong L[X_1,\ldots ,X_v]/I_0$. Suppose the ideal $I_0$ is generated by power products
of degree 2. Then for every residually rational flat local homomorphism $(A,\hbox{\eufm m})\to
(B,\hbox{\eufm n})$ of local $L$-algebras that has a special fiber isomorphic to $B_0$ the
$(v+1)$th sum transforms of the local Hilbert series of $A$ and $B$ satisfy the Lech inequality
$H_A^{v+1}\le H_B^{v+1}$. \item{[H]} Herzog, Bernd: {\it Kodaira-{S}pencer maps in local algebra}.
Lecture Notes in Mathematics 1597. Springer-Verlag, Berlin Heidelberg 1994.
- The Wallace lines of a triangle in the affine-metric plane over R were studied by O.
Giering [Journal for Geometry and Graphics 1 (1997) 119-133]. This paper deals with the
isotropic or galilean case [see E. M. Schröder, Vorlesungen über Geometrie. Bd. 3: Metrische
Geometrie. BI-Verlag, Mannheim 1992] - which is not included in the paper of Giering. Essential
means is the d-footpoint definition of J. Lang [Zur isotropen
Dreiecksgeometrie und zum Appolonischen Berührproblem in der isotropen Ebene, Ber. Math.-Stat.
Sekt. Forschungszentrum Graz 20 (1983) 1-11].
- It was conjectured that the smallest minimal point sets of PG(2s, q), q a square,
that meet every s-subspace and that generate the whole space are Baer subgeometries
PG(2s, Squareroot of q). This was shown in 1971 by Bruen for s = 1, and by K. Metsch
and L. Storme [Beitr. Algebra Geom., submitted] for s =2. Our main interest in this
paper is to prepare a possible proof of this conjecture by proving a strong theorem
on line-blocking sets in projective spaces (see Theorem 1.1). We apply this theorem
to prove the conjecture in the case s = 3. The general case will be handled in a
forthcoming paper by the first author.
- We consider rings S, not necessarily with 1, and develop a decomposition theory
for submonoids and subgroups of (S, circle) where the circle operation "circle" is
defined by x circle y = x + y - xy. Decompositions are expressed in terms of internal
semidirect, reverse semidirect and general products, which may be realised externally
in terms of naturally occurring representations and antirepresentations. The theory is
applied to matrix rings over S when S is radical, obtaining group presentations in terms
of (S, +) and (S, circle). Further details are worked out in special cases when
S = p Zpt for p prime and t >= 3.
- We characterize isomorphisms of generalized polygons (in particular automorphisms) by maps
on flags which preserve a certain fixed distance. In Part II, we consider maps on point and/or
lines. Exceptions give rise to interesting properties, which on their turn have some nice
applications.
- For a right module M over a ring R, let S be the endomorphism ring of the
module M. The left annihilator of a submodule A of M in S is denoted l(A).
Extending the notion of an Ikeda-Nakayama ring, we call the module M an
Ikeda-Nakayama module if, for any two submodules A and B of M, the left
annihilator of the intersection of A and B in S is equal to l(A) + l(B).
Various properties of these modules are derived and then applied to
quasi-continuous rings, dual rings and QF-rings to obtain several related
results.
- We discuss a unified way to derive the convex semiregular polyhedra from the Platonic
solids. Based on this we prove that, among the Archimedean solids, Cubus simus (i.e., the
snub cube) and Dodecaedron simum (the snub dodecahedron) can be characterized by the following
property: it is impossible to construct an edge from the given diameter of the circumsphere
by ruler and compass.
- In plane equiaffine differential geometry an oval is called "affinely convex" iff
every hyperosculating parabola of the oval supports it. Such curves may be characterized
by nonnegative affine curvature. As a consequence of the theorems of C. Perry the affine
perimeter is strictly monotone increasing and continuous for affinely convex ovals. We
prove generalizations for affinely convex parabola polygons and affinely convex ovaloids
in higher dimensions.
- This paper deals with the characterization of the sums of compact convex sets with
linear subspaces, simplices, sandwiches (convex hulls of pairs of parallel affine
manifolds) and parallelotopes in terms of the so-called internal and conical
representations, topological and geometrical properties. In particular, it is
shown that a closed convex set is a sandwich if and only if its relative boundary
is unconnected. The characterizations of families of closed convex sets can be useful
in different fields of applied mathematics. For instance, it is proved that a bounded
linear semi-infinite programming problem whose feasible set is the sum of a compact convex
set with a linear subspace is necessarily solvable and has zero duality gap.
- The existence of mixed curvature measures of two sets in Rd with positive reach
introduced in a previous paper of the authors [Mixed curvature measures for sets of positive
reach and a translative integral formula, Geom. Dedicata 57 (1995) 259-283] is discussed. An
example shows that the non-osculating condition from the cited paper does not ensure the
locally bounded variation of the mixed curvature measures. Further, some sufficient conditions
for the local boundedness of mixed curvature measures involving absolute curvature measures are
presented.
- We study the 3-dimensional contact metric manifolds equipped with constant
x-sectional curvature and f-sectional
curvature or constant norm of the Ricci operator.
- We associate a 3-manifold to a multiple valued semi-conformal mapping on the 3-sphere
and study the branching set of the corresponding covering projection. Some remarkable
geometric structures occur.
- The aim of this paper is to study the Weierstrass semigroup of ramified points on
non-singular models for curves on a rational normal scroll. We find properties of this
semigroup and determine it in some special cases, finding also a geometrical interpretation
for some of the Weierstrass gaps.
- We introduce a method of invariant decompositions of tensor spaces over the
real, finite-dimensional vector spaces. The method is elementary, and is
based on the concept of a natural projector (i.e., the projector eqiuvariant
with respect to the general linear group). A new, unified invariant
decomposition theory is obtained, including as special cases the classical
Young-Kronecker decompositions of spaces of covariant (resp. contravariant)
tensors, as well as the decompositions by the trace operation. As an example
we find the complete list of natural projectors in the space of tensors of
type (1,2). We show, in particular, that there exist families of natural
projectors, depending on real parameters, defining new representations of
the general linear group in certain vector subspaces.
- We give characterizations of the property of being weakly reductive for finitely generated
commutative semigroups. As a consequence of these characterizations we obtain an algorithm for
determining from a presentation of a finitely generated commutative semigroup whether it is
weakly reductive. Furthermore, we present some connections between cancellative finitely
generated commutative semigroups, Archimedean weakly reductive finitely generated commutative
semigroups and N-semigroups.
- The concept of perfection of a polytope was introduced by S. A. Robertson. Intuitively
speaking, a polytope P is perfect if and only if it cannot be deformed to a polytope of
different shape without changing the action of its symmetry group G(P) on its face-lattice
F(P). By Rostami's conjecture, the perfect 4-polytopes form a particular set of Wythoffian
polytopes. In the present paper first this known set is briefly surveyed. In the rest of
the paper 2 new classes of perfect 4-polytopes are constructed and discussed, hence
Rostami's conjecture is disproved. It is emphasized that in contrast to an existing opinion
in the literature, the classification of perfect 4-polytopes is not complete as yet.
- This paper is related to the third author's previous result on the existence of volume
polynomials for a given polyhedron having only triangular faces. We simplify his original
proof in the case when the polyhedron is homeomorphic to the 2-sphere. Our approach exploits
the fact that any such polyhedron contains a so-called clean vertex - that is, a vertex not
incident with any nonfacial cycle composed of 3 edges. This fact appears as one of the main
results of the article. Also, we characterize triangulations reducible to a tetrahedron by
repeatedly removing 3-valent vertices, and estimate the degree of volume polynomials. We
address the torus case too.
- We consider the traceless part C-tilde of the difference tensor field C between the
Levi-Civita connections of the first and the third fundamental forms for non-degenerate
surface immersions in S3(1). In analogy to affine differential geometry of
Rn+1 where quadrics are characterized by the vanishing of a traceless cubic
form, we study the condition C-tilde equivalent 0, give examples and classify
non-degenerate surfaces in S3(1) which satisfy this condition.
- It is shown that a piecewise linear function on a convex domain in Rd
can be represented as a boolean polynomial in terms of its linear components.
- We characterize isomorphisms of generalized polygons (in particular automorphisms)
by maps on points and/or lines which preserve a certain fixed distance. In Part I, we
considered maps on flags. Exceptions give rise to interesting properties, which on their
turn have some nice applications.
- [Abstract-pdf]
Let $L_1$ be a finite simple sloop of cardinality $n$ or the 8-element sloop.
We construct a subdirectly irreducible (monolithic) sloop $L=2\otimes_\alpha L_1$
of cardinality $2n$, for each $n\geq 8$, with $n\equiv 2$ or 4 (mod 6), in which
each proper homomorphic image is a Boolean sloop. R. W. Quackenbush [{\it Varieties
of Steiner loops and Steiner quasigroups}, Canad. J. Math. 18 (1978) 1187-1198]
has proved that the variety $V( L_1)$ generated by a finite simple planar sloop
$L_1$ covers the smallest non-trivial subvariety (the class of all Boolean sloops).
For any finite planar sloop $L_1$, the variety $V( L)$ generated by the constructed
sloop $L= 2 \otimes_\alpha L_1$ covers the variety $V( L_1)$.
- Let R be a prime ring. It is shown that, under certain restrictions on char(R),
R admits a functional identity f(x)xf(x) ... xf(x) = 0, x from R, where f from R
to R is a nonzero additive map, if and only if its central closure S contains an
idempotent e not equal to 0 or 1 such that eSe = Ce where C is the extended
centroid of R.
- We solve problems of Buffon type for an arbitrary convex body of revolution and
four different types of lattices.
- [Abstract-pdf]
Let $k$ be a field, $R$ a $k$-algebra and $A=R[\theta_1, \theta_2,\ldots,\theta_n]$ a
Poincaré-Birkhoff-Witt extension of $R$. If each $\theta_i$ acts locally finitely on $R$,
then we show that $GKdim(A)=GKdim(R) +n$. From this we deduce some results concerning
incomparability, prime length and Tauvel's height formula in the crossed product
$R\star g$ where $g$ is a finite-dimensional Lie algebra acting as derivations on $R$.
Similar results are obtained for $R\star G$, where $G$ is a free abelian group of finite
rank. As a corollary of the results for $G$, Tauvel's height formula is established in
$R\otimes _kP(\lambda)$ where $P(\lambda)$ is the quantum torus.
- Suppose S is a finite generalized quadrangle (GQ) of order (s, t), s and t not equal to
1, and suppose that L is a line of S. A symmetry about L is an automorphism of the GQ which
fixes every line of S meeting L (including L). A line is called an axis of symmetry if there
is a full group of symmetries of size s about this line, and a point of a generalized quadrangle
is a translation point if every line through it is an axis of symmetry. A GQ with a translation
point is often called a translation generalized quadrangle. In the present paper, we classify
the generalized quadrangles with at least two distinct translation points. In order to obtain
the main result, we prove many more general theorems which are useful for the theory of
span-symmetric generalized quadrangles (these are the GQ's with non-concurrent axes of
symmetry), and using earlier results of the author, we give more general versions of our
main theorem.
As a by-product of the proof of our main result, we will show that for any span-symmetric
generalized quadrangle S of order (s, t), s and t not equal to 1, s and t are powers of the
same prime, and if s not equal to t and s is odd, then S always contains at least s + 1
classical subquadrangles of order s.
In an addendum we obtain an explicit construction of some classes of spreads for the
point-line duals of the Kantor flock generalized quadrangles as a second by-product of
the proof of our main result.
- [Abstract-pdf]
We introduce a new computational technique for $n\times n$ matrices, over a
$\Bbb{Z}_{2}$-graded ring $R=R_{0}\oplus R_{1}$ with $R_{0}\subseteq Z(R)$, leading
us to a new concept of determinant, which can be used to derive an invariant
Cayley-Hamilton identity. An explicit construction of the inverse matrix $A^{-1}$
for any invertible $n\times n$ matrix $A$ over a Grassmann algebra $E$ is also obtained.
- For someone familiar with the notion of self-parallel group for an immersion
into euclidean space [FR], [W] it is only natural to wonder what happens if, in
the case of space curves, normal planes are replaced by, say, osculating planes.
We give here a necessary and sufficient condition for the non-triviality of the
osculating group of a simple space curve. This is, in the new situation, the group
corresponding to the self-parallel group. Problems of a similar nature have also
been considered in [BCW] and [CR].
[BCW] d'Azevedo Breda, A. M.; Craveiro de Carvalho, F. J.; Wegner, Bernd: On the
existence of Bertrand pairs. Pré-publiccao 01-06, Departamento de Matematica da
Universidade de Coimbra.
[CR] Craveiro de Carvalho, F. J.; Robertson, S. A.: The parallel group of a plane
curve. Proceedings of the First International Meeting on Geometry and Topology,
Braga (1997) 57-61.
[FR] Farran, H. R.; Robertson, S. A.: Parallel immersions in euclidean space. J.
London Math. Soc. 35 (1987) 527-538.
[W] Wegner, B.: Self-parallel and transnormal curves. Geom. Dedicata 38 (1991)
175-191.
- Every sequence of squares with total area greater than or equal to 2 permits a
covering of the unit square.
- We give a complete description of subdirectly irreducible rings with involution
satisfying xn+1 = x for some positive integer n. We also discuss ways to
apply this result for constructing lattices of varieties of rings with involution
obeying an identity of the given type.
- Recently J. Fukuta [Amer. Math. Monthly 103 (1996) 267; Math. Mag. 69 (1996) 67;
Math. Mag. 70 (1997) 70--73] and Z. Cerin [Rad. Mat. 9 (1999) 227--239] showed how
regular hexagons can be associated to any triangle, thus extending Napoleon's theorem.
The aim of this paper is to prove that these results are closely related to linear
maps. This reflects better the affine character of some constructions and gives also
rise to a few new theorems.
- It is well known that optimal ternary codes of length 5 and covering radius
1 have 27 codewords. The structure of such optimal codes has been studied, but a
classification of these is still lacking. In this work, a complete classification
is carried out by constructing codes coordinate by coordinate in a backtrack search.
Linear inequalities and equivalence checking are used to prune the search. In total
there are 17 optimal codes.
- Quadratically parametrized smooth maps from one complex projective space to
another are constructed as projections of the Segre map of the complexification.
A classification theorem relates equivalence classes of projections to congruence
classes of matrix pencils. Maps from the 2-sphere to the complex projective plane,
which generalize stereographic projection, and immersions of the complex projective
plane in four and five complex dimensions, are considered in detail. Of particular
interest are the CR singular points in the image.
- We investigate similarities between the category of vector spaces and that of
polytopal algebras, containing the former as a full subcategory. In Section 2 we
introduce the notion of a polytopal Picard group and show that it is trivial for
fields. The coincidence of this group with the ordinary Picard group for general
rings remains an open question. In Section 3 we survey some of the previous results
on the automorphism groups and retractions. These results support a general conjecture
proposed in Section 4 about the nature of arbitrary homomorphisms of polytopal algebras.
Thereafter a further confirmation of this conjecture is presented by homomorphisms
defined on Veronese singularities.
This is a continuation of the project started in [BG1], [BG2], [BG3]. The higher
K-theoretic aspects of polytopal linear objects will be treated in [BG4], [BG5].
[BG1] Bruns, W.; Gubeladze, J.: Polytopal linear groups, J. Algebra 218 (1999) 715-737.
[BG2] Bruns, W.; Gubeladze, J.: Polytopal linear retractions, Trans. Amer. Math. Soc.
354(1) (2002) 179-203.
[BG3] Bruns, W.; Gubeladze, J.: Polyhedral algebras, arrangements of toric varieties,
and their groups, Adv. Stud. Pure Math., to appear.
[BG4] Bruns, W.; Gubeladze, J.: Polyhedral K2 , preprint.
[BG5] Bruns, W.; Gubeladze, J.: Higher polyhedral K-groups , preprint.
- For a convex body K, let us denote by t(K) the largest number for which there
exists a packing with finitely many translates of K in which every translate has
at least t(K) neighbours. In this paper we determine t(K) for convex discs and
3-dimensional convex cylinders. We also examine how small the cardinalities of
the extremal configurations can be in these cases.
- A metabelian group G of order 1440 is constructed which provides a counterexample
to a conjecture of Zassenhaus on automorphisms of integral group rings. The group is
constructed in the spirit of K. W. Roggenkamp and L. L. Scott [On a conjecture of
Zassenhaus for finite group rings, Manuscript, November 1988, 1-60]. An augmented
automorphism of ZG , Z standing for the integers, which has no Zassenhaus factorization
is given explicitly (this was already done by L. Klingler [Construction of a counterexample
to a conjecture of Zassenhaus, Commun. Algebra 19 (1991) 2303-2330] for a group of order
6720), but this time only a few distinguished group ring elements are used for its
construction, carefully exploiting certain congruence relations satisfied by powers of
these elements.
- [Abstract-pdf]
We consider a class of real hypersurfaces of the Grassmann manifold of $k$-planes in
${\mathbb C}^{n}$, $G_{k}({\mathbb C}^{n})$, for $k > 2$. Namely the family of tubes around
$G_{k}({\mathbb C}^{m})$ with $m < n$ and around the quaternionic Grassmann manifold of
$k/2$-quaternionic planes in ${\mathbb H}^{n/2}$, $G_{k/2}({\mathbb H}^{n/2})$, when $k$
and $n$ are even. We determine which of those tubes are homogeneous and for them we find
the spectral decomposition of the shape operator. As a consequence we show that they are
Hopf hypersurfaces.
- [Abstract-pdf]
We present efficient algorithms for the on-line $q$-adic covering of the unit interval
by sequences of segments. The basic method guarantees covering provided the total length
of segments is at least $1+ 2\cdot {1\over q} - {1\over q^3}$. Other algorithms improve
this estimate for $q\geq 6$. The unit $d$-dimensional cube can be on-line covered by an
arbitrary sequence of cubes whose total volume is at least $2^d+{5\over 3}+{5\over 3}\cdot
2^{-d}$.
- Based on Weitzenböck's theorem and Nagata's counterexample for Hilbert's fourteenth
problem we construct two finitely generated invariant rings R ans S, where S is a subset
of K[x1, x2, ... , xn] s.t. the intersection of R and
S is not finitely generated as a K-algebra.
- [Abstract-pdf]
This paper provides a proof the following result: Suppose $n\geq 2$,
$0 < c < s$, $0 < c/s < \frac{\sqrt{5}-1}{2}$, and $f\colon
E^{n}\to E^{m}$ is any function such that for all $p,q\in E^{n}$, the
following two properties hold: (1) If $|p-q|=c$, then $|f(p)-f(q)|\leq c$.
(2) If $|p-q|=s$, then $|f(p)-f(q)|\geq s$. Then $f$ is congruence. This
result originally proved Bezdek and Connelly [BC]. However Bezdek and
Connelly's result depends on a result of Rado et al. [R], where the similar
result holds but for $ \frac{1}{\sqrt{3}} $ replacing $\frac{\sqrt{5}-1}{2}$.
Rado's proof is long and time consuming to verify.
In this paper, we provide our own separate proof, using the mathematical software,
Maple, which is independent of Rado's proof. Our result is shorter, more systematic
and, we believe, easier to verify.
[BC] Bezdek, K.; Connelly, R.: Two-Distance Preserving Functions From Euclidean Space,
Period. Math. Hungar. 39(1-3) (1999) 185-200.
[R] Rado, F.; Andreescu, D.; Valcan, C.: Mapping of $E^{n}$ to $E^{m}$ preserving
two distance, Seminar on Geometry (Cluj-Napoca) 9-22, 1986.
- We examine subalgebras on two generators in the univariate polynomial ring.
A set, S, of polynomials in a subalgebra of a polynomial ring is called a canonical
basis (also referred to as SAGBI basis) for the subalgebra if all lead monomials in
the subalgebra are products of lead monomials of polynomials in S. In this paper we
prove that a pair of polynomials {f, g} is a canonical basis for the subalgebra they
generate if and only if both f and g can be written as compositions of polynomials
with the same inner polynomial h for some h of degree equal to the greatest common
divisor of the degrees of f and g. Especially polynomials of relatively prime degrees
constitute a canonical basis. Another special case occurs when the degree of g is a
multiple of the degree of f. In this case {f, g} is a canonical basis if and only if
g is a polynomial in f.
- Let R be a polynomial ring over a field of dimension n. Let
m be the maximal homogeneous ideal of R. Let I be a complete
intersection homogeneous ideal of R minimally generated by
polynomials of degrees d1, d2, ... ,
dn. F. S. Macaulay showed that for all integers
i = 0, 1, ..., d + 1, I : mi
= I + md + 1 -
i where d = d1
+ d2 + ... + dn - n . We provide a modern
proof of a generalization of this result to standard graded
Gorenstein rings.
- In a recent paper of the first author [Polyhedral maps with few edges, Topics
in Comb. and Graph Theory (Ringel-Festschrift) (eds. Bodendiek, R. and Henn, R.),
Physica-Verlag, Heidelberg 1990, 153-162], a {5, 5}-equivelar polyhedral map of
Euler characteristic -8 was constructed. In this article we prove that {5, 5}-equivelar
polyhedral map of Euler characteristic -8 is unique. As a consequence, we get that the
minimum number of edges in a non-orientable polyhedral map of Euler characteristic -8
is > 40. We have also constructed {5, 5}-equivelar polyhedral map of Euler
characteristic -2m for each m >= 4.
- \font\fr=eufm10 \font\bb=msbm10 The group of automorphisms of the coordinate ring of quantum
symplectic space ${\cal O}_q(\hbox{\fr sp\bb\char'174}^{2\times n})$ is isomorphic to the algebraic
torus $(\hbox{\bb\char'174}^\times)^{n+1}$ when $q$ is not a root of unity.