Äquivalenz und Isomorphie bei Bewertungsfortsetzungen auf einfach-transzendente Erweiterungen

It is well known that there are three types of prolongations of valuations from a field K to , t a transcendent over K. In this paper, we give a representation of the equivalence and isomorphy classes of each type with the help of an algebraic closure of K. Furthermore, upper and lower bounds for the cardinalities of these classes are presented, and we answer the question regarding when a subfield of allows a classification of the prolongation of valuations to .

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Received: 18 November 1998; in final form: 9 May 2000 / Published online: 17 May 2001     

Free arrangements and rhombic tilings

Let Z be a centrally symmetric polygon with integer side lengths. We answer the following two questions:

1. When is the associated discriminantal hyperplane arrangementfree in the sense of Saito and Terao?
2. When areall of the tilings of Z by unit rhombicoherent in the sense of Billera and Sturmfels?

Surprisingly, the answers to these two questions are very similar. Furthermore, by means of an old result of MacMahon on plane partitions and some new results of Elnitsky on rhombic tilings, the answer to the first question helps to answer the second. These results then also give rise to some interesting geometric corollaries. Consideration of the discriminantal arrangements for some particular octagons leads to a previously announced counterexample to the conjecture by Saito [ER2] that the complexified complement of a real free arrangement is aK (π, 1) space.     

On the number of points of bounded height¶on arithmetic projective spaces

Let K be a number field and its ring of integers. Let be a Hermitian vector bundle over . In the first part of this paper we estimate the number of points of bounded height in (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d>1 in of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of ℙ ^N _K blown up at a linear subspace of codimension two. Received: 20 February 1998 / Revised version: 9 November 1998     

On a spherical integral transformation and sections of star bodies

LetK be ad-dimensional star body (with respect to the origino). It is known that the (d–1)-dimensional volume of the intersections ofK with the hyperplanes througho does not uniquely determineK. Uniqueness can only be achieved under additional assumptions, such as central symmetry. Here it is pointed out that if one uses, instead of intersections by hyperplanes, intersections by half-planes that containo on the boundary, then, without any additional assumptions, the volume of these intersections determinesK uniquely. This assertion, and more general results of this kind, together with stability estimates, are obtained from uniqueness results and estimates concerning a particular spherical integral transformation.Supported by National Science Foundation Research Grant DMS-9401487     

RESTRICTING THE COMPARISON FUNCTOR OF AN ADJUNCTION TO PROJECTIVE OBJECTS

Abstract

This paper deals with projectives (in the sense of K.A.Hardie [5] relative to a right adjoint functor U: A → K. We answer the question, raised by R.-E. Hoffmann [6] p. 135, of knowing under what conditions there exists an equivalence between Proj u and Proj U^r, induced by the comparison functor Φ: A → K^T, where T denotes the monad induced by U. In the case, that U is an algebraic functor we also give necessary and sufficient conditions for the re gular projective objects to coincide with the U-projectives. Finally, we delineate how these results nay be applied in certain familiar situations.     

A characterization of the<Emphasis Type="Italic">MM</Emphasis>*-position of a convex body in terms of covariance matrices

We characterize the position of a convex bodyK such that it minimizesM(TK)M*(TK) (theMM*-position) in terms of properties of the measures ‖ · ‖ _K d σ(·) and ‖ · ‖ _K °d σ(·), answering a question posed by A. Giannopoulos and V. Milman. The techniques used allow us to study other extremal problems in the context of dual Brunn-Minkowski theory. Partially supported by MCYT Grant (Spain) BFM2001-1793 and FEDER funds from UE. Partially supported by MCYT Grant (Spain) BFM2001-1793 and FEDER funds from UE.     

On the minimal genus of 2-complexes

Gross and Rosen asked if the genus of a 2-dimensional complex K embeddable in some (orientable) surface is equal to the genus of the graph of appropriate barycentric subdivision of K. We answer the nonorientable genus and the Euler genus versions of Gross and Rosen's question in affirmative. We show that this is not the case for the orientable genus by proving that taking ⌊ log₂ g⌋ th barycentric subdivision is not sufficient, where g is the genus of K. On the other hand, (1+⌈log₂(g+2)⌉)th subdivision is proved to be sufficient. © 1997 John Wiley & Sons, Inc.     

Über relativ-lnvariante Kreiseinheiten und Stickelberger-Elemente

We consider an extensionE ?K of absolutely abellan number fields and the corresponding groups of circular unitsC _E ?C _K in the sense of Sinnott. In this paper we consider the question: Is every Gal(K/E)-invarlant element ofC _K already inC _E ? This has been answered in the affirmative recently by Gold and Kirn in the case that bothE andK are cyclotomlc fields. We show that the question has an affirmative answer ifK is cyclic over ?, and that the answer in general is negative. There is an analogous question concerning Stickelberger ideals (the inclusion map now being replaced by the corestriction map), and the answer to that question is shown to be exactly the same as to the first one.     

Values of rational functions on non-hilbertian fields and a question of Weissauer

We answer in the negative a question raised by Fried and Jarden, asking whether the quotient field of a unique factorization domain with infinitely many primes is necessarily hilbertian. This implies a negative answer to a related question of Weissauer. Our constructions are simple and take place inside the field of algebraic numbers. Simultaneously we investigate the relation of hilbertianity of a fieldK with the structure of the value sets of rational functions onK: we construct a non-hilbertian subfieldK of such that, given anyf ₁ ,…,f _h ∈K(x), each of degree ≥2, the union ∪ _z=1 ^h f _z(K) does not containK. See e.g. [FrJ], [L1], [L2], [Sch], [Se1], or [Se2] for the classical theory of hilbertian fields.     

On certain problems in the analytical arithmetic of quadratic forms arising from the theory of curves of genus 2 with elliptic differentials

For an imaginary quadratic fieldK we study the asymptotic behaviour (with respect top) of the number of integers inK with norm of the formk(p−k) for some 1≤k≤p−1, wherep is a prime number. The motivation for studying this problem is that it is known by recent results due to G. Frey and E. Kani that knowledge of this asymptotic behaviour can lead to statements of existence of curves of genus 2 with elliptic differentials in particular cases. We give a general, and from one point of view complete, answer to this question on asymptotic behaviour. This answer is derived from a theorem concerning the number of representations of a natural number by certain quaternary quadratic forms. This second result may be of some independent interest because it can be seen as a generalisation of the classical theorem of Jacobi on the number of representations of a natural number as a sum of 4 squares.     

Horospheres and Convex Bodies in n-Dimensional Hyperbolic Space

In n-dimensional Euclidean space, the measure of hyperplanes intersecting a convex domain is proportional to the (n–2)-mean curvature integral of its boundary. This question was considered by Santaló in hyperbolic space. In non-Euclidean geometry the totally geodesic hypersurfaces are not always the best analogue to linear hyperplanes. In some situations horospheres play the role of Euclidean hyperplanes.In dimensions n=2 and 3, Santaló proved that the measure of horospheres intersecting a convex domain is also proportional to the (n–2)-mean curvature integral of its boundary.In this paper we show that this analogy does not generalize to higher dimensions. We express the measure of horospheres intersecting a convex body as a linear combination of the mean curvature integrals of its boundary.     

Groups with trivial virtual automorphism group

We answer a question of A. Lubotzky and A. Mann by constructing examples of infinite groupsG such that every isomorphismα:H →K between subgroupsH andK having finite index inG coincides with the identity on some subgroup of finite index. The structure of such a group is very restricted;G must be virtually a 2-group with finite central derived subgroup andG/G′ elementary abelian. This work was begun while the second author was a visitor at the University of Padova. He wishes to thank the Mathematics Department for its hospitality and the C.N.R. for its financial support.     

On Banach spaces whose norm-open sets are -sets in the weak topology

We investigate non-separable Banach spaces whose norm-open sets are countable unions of sets closed in the weak topology and a narrower class of Banach spaces with a network for the norm topology which is σ-discrete in the weak topology. In particular, we answer a question of Arhangel'skii exhibiting various examples of non-separable function spaces C(K) with a σ-discrete network for the pointwise topology and (consistently) we answer some questions of Edgar and Oncina concerning Borel structures and Kadec renormings in Banach spaces.     

On the Independence of Hartman Sequences

 Given and , we define by setting if and only if , where denotes the fractional part of α, i.e. α is considered as an element of the torus . If the topological boundary of A has Haar measure 0, then is called a Hartman sequence, which is a generalisation of Kronecker and Beatty sequences. In this article we answer a question of Winkler by showing explicitly for which sets , and vectors , we have . The main tool of the proof is Weyl’s theorem on uniform distribution. Received 3 November 2000; in final form 24 April 2001     

On the identities of the Grassmann algebras in characteristicp>0

In this note we exhibit bases of the polynomial identities satisfied by the Grassmann algebras over a field of positive characteristic. This allows us to answer the following question of Kemer: Does the infinite dimensional Grassmann algebra with 1, over an infinite fieldK of characteristic 3, satisfy all identities of the algebraM ₂(K) of all 2×2 matrices overK? We give a negative answer to this question. Further, we show that certain finite dimensional Grassmann algebras do give a positive answer to Kemer's question. All this allows us to obtain some information about the identities satisfied by the algebraM ₂(K) over an infinite fieldK of positive odd characteristic, and to conjecture bases of theidentities ofM ₂(K).     

Coupling vs. conductance for the Jerrum–Sinclair chain*

We address the following question: is the causal coupling method as strong as the conductance method in showing rapid mixing of Markov chains? A causal coupling is a coupling which uses only past and present information, but not information about the future. We answer the above question in the negative by showing that there exists a bipartite graph G such that any causal coupling argument on the Jerrum–Sinclair Markov chain for sampling almost uniformly from the set of perfect and near perfect matchings of G must necessarily take time exponential in the number of vertices in G. In contrast, the above Markov chain on G has been shown to mix in polynomial time using conductance arguments. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 1–17, 2001     

Unique determination of convex polytopes by non-central sections

A question of Barker and Larman asks whether convex bodies that contain a sphere of radius t in their interiors are uniquely determined by the volumes of sections by hyperplanes tangent to the sphere. We affirmatively solve this problem for convex polytopes.     

A geometric connection to threshold logic via cubical lattices

A cut-complex is a cubical complex whose vertices are strictly separable from the rest of the vertices of the n-cube by a hyperplane of R ⁿ . These objects render geometric presentations for threshold Boolean functions, the core objects of study in threshold logic. By applying cubical lattices and geometry of rotating hyperplanes, we prove necessary and sufficient conditions to recognize the cut-complexes with 2 or 3 maximal faces. This result confirms a positive answer to an old conjecture of Metropolis-Rota concerning cubical lattices.     

Structure of maximal commutative subalgebras of the first Weyl Algebra

A short proof is given to Dixmier's sixth problem from [9] for the first Weyl algebra which askswhether a polynomial of a generic element of the first Weyl algebra is a generic element. Anaffirmative answer to this question was given by A. Joseph [14]. In the paper we give an answer to a similar question but for an arbitrary element of the first Weyl algebra. This result is used then to clarify structure of maximal commutative subalgebras in the first Weyl algebraA ₁: for a given maximal commutative subalgebra C of the Weyl algebra A₁ (almost) all non-scalar elements of it have the sametype, more precisely, have one of the following types: (i) strongly nilpotent, (ii) weakly nilpotent, (iii) generic, (iv) generic except for a subset, K ^*a+K of elements of strongly semi-simple type where a∈C is an element of strongly semi-simple type and K^*=K/{0}, (v) generic except for a subset, K^*a+K of elements of weakly semi-simple type where a∈C is an element of weakly semi-simple type.

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Sunto Si fornisce una dimostrazione breve al sesto problema di Dixmier enunciato in [9] per la prima algebra di Weyl che chiedese un polinomio di un elemento generico della prima algebra di Weyl è un elemento generico. Una rispostaaffermativa a questo problema à stata data da A. Joseph in [14]. In questo articolo formiamo una risposta ad un quesito simile ma per un elemento arbitrario della prima algebra di Weyl. Questo risultato è usato quindi per chiarire la struttura delle sottoalgebre commutative massimali della prima algebra di WeylA ₁: per una data sottoalgebra commutativa massimale C dell'algebra di Weyl A₁ (quasi) tutti i suoi elementi non scalari hanno lo stessotipo; più precisamente, hanno uno dei seguenti tipi: (i) fortemente nilpotente, (ii) debolmente nilpotente, (iii) generico, (iv) generico eccetto che per un sottoinsieme K ^*a+K di elementi di tipo fortemente semisemplice, dove a∈C è un elemento di tipo debolmente semisemplice.

     

On the (U+K)-orbits of certain weighted shifts

D. A. Herrero has defined the (U+K)-orbit of an operatorT acting on an Hilbert space as (U+K)(T)={R ^–1 TR: R invertible of the form unitary plus compact}. In this paper we consider the (U+K)-orbit and the closure thereof for bilateral and unilateral weighted shifts. In particular, we determine which shifts are in the (U+K)-orbits of injective weighted shifts and which shifts are in the closure of the (U+K)-orbit of periodic injective shifts. Also, the closure of the (U+K)-orbit of injective essentially normal shifts is determined.Research supported in part by an NSERC operating grant.