Sums of lengtht in Abelian groups

LetG be an Abelian group written additively,B a finite subset ofG, and lett be a positive integer. Fort≦|B|, letB _t denote the set of sums oft distinct elements overB. Furthermore, letK be a subgroup ofG and let σ denote the canonical homomorphism σ:G→G/K. WriteB _t (modB _t) forB _tσ and writeB _t (modK) forBσ. The following addition theorem in groups is proved. LetG be an Abelian group with no 2-torsion and letB a be finite subset ofG. Ift is a positive integer such thatt<|B| then |B _t (modK)|≧|B (modK)| for any finite subgroupK ofG.     

Some Elements of Finite Order in K2Q

Let K2 be the Milnor functor and let Фn （x）∈ Q[X] be the n-th cyclotomic polynomial. Let Gn（Q） denote a subset consisting of elements of the form {a, Фn（a）}, where a ∈ Q^＊ and {, } denotes the Steinberg symbol in K2Q. J. Browkin proved that Gn（Q） is a subgroup of K2Q if n = 1,2, 3, 4 or 6 and conjectured that Gn（Q） is not a group for any other values of n. This conjecture was confirmed for n =2^T 3S or n = p^r, where p ≥ 5 is a prime number such that h（Q（ζp）） is not divisible by p. In this paper we confirm the conjecture for some n, where n is not of the above forms, more precisely, for n = 15, 21,33, 35, 60 or 105.     

Tensor products and dimensions of simple modules for symmetric groups

LetK be an algebraically closed field of characteristic,p>0 and letD ^λ be the simple modules of the symmetric groupS _r overK where λ is a p-regular partition ofr. The dimensions ofD ^λ for λ with at mostn parts are the same as the multiplicities of direct summands ofD ^⊗r whereE is the natural module for the groupGL _n (K). Whenn=2 we determine generating functions for these multiplicities and hence for the dimensions ofD ^λ for all partitions λ with two parts. These can be expressed as rational functions of Chebyshev polynomials; and we obtain explicit formulae for the coefficients.     

On the essential dimension of cyclic <Emphasis Type="Italic">p</Emphasis>-groups

Let p be a prime number, let K be a field of characteristic not p, containing the p-th roots of unity, and let r≥1 be an integer. We compute the essential dimension of ℤ/p ^r ℤ over K (Theorem 4.1). In particular, i) We have ed_ℚ(ℤ/8ℤ)=4, a result which was conjectured by Buhler and Reichstein in 1995 (unpublished). ii) We have ed_ℚ(ℤ/p ^r ℤ)≥p ^r-1.     

On galois groups and their maximal 2-subgroups

LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL ₁ of ℚ(T) with Galois groupG. LetL be the subfield ofL ₁ fixed byH. We make the hypothesis thatL ₁ admits a quadratic extensionL ₂ which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL ₁ then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL ₂. IfG is a 2-group, we show that there exists an element α ε ℚ(T) such thatL ₁ admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions of ℚ(T).     

Free pro-p groups as galois groups over ℚ(p)(t)

Letp be a prime and let ℚ(p) denote the maximalp-extension of ℚ. We prove that for every primep, the free pro-p group on countably many generators is realizable as a regular extension of ℚ(p)(t). As a consequence, if ℚ _nil denotes the maximal nilpotent extension of ℚ, then every finite nilpotent group is realizable as a regular extension of ℚ _nil (t).     

Galois groups ofp-extensions with two ramification points

LetK _p (p, q) be the maximalp-extension of the field ℚ of rational numbers with ramification pointsp andq. LetG _p (p, q) be the Galois group of the extensionK _p(p.q)/ℚ. It is known thatG _p(p, q) can be presented by two generators which satisfy a single relation. The form of this relation is known only modulo the second member of the descending central series ofG _p(p, q). In this paper, we find an arithmetical-type condition on which the form of the relation modulo the third member of the descending central series ofG _p(p, q) depends. We also consider two examples withp=3,q=19 andp=3,q=37. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 1, pp. 48–60, January–March, 2000. Translated by H. Markšaitis     

Coboundaries of irreducible markov operators onC(K)

LetK be a compact Hausdorff space, and letT be an irreducible Markov operator onC(K). We show that ifgεC(K) satisfies sup _N ‖Σ _j ^=0N T ^j g‖<∞, then (and only then) there existsfεC(K) with (I − T)f=g. Generalizing the result to irreducible Markov operator representations of certain semi-groups, we obtain that bounded cocycles are (continuous) coboundaries. For minimal semi-group actions inC(K), no restriction on the semi-group is needed.     

Spectral mapping theorems for exponentially bounded c-semigroups in Banach spaces

LetZ be a generator of an exponentially boundedC-semigroup {S _t } _t≥0 in a Banach space and letT _t =C ⁻¹ S _t . We show that the spectral mapping theorems such as exp(tσ(Z)) ⊂ σ(T _t ) and exp(tσ _p (Z)) ⊂ tσ _p (T _t ) ⊂ exp(tσ _p (Z)) ⋃ {0} for everyt≥0 hold. The present studies were supported by the Basic Science Research Institute Program, Ministry of Education, 1987.     

Riemann surfaces in fibered polynomial hulls

Let Δ be the closed unit disk in C, let Γ be the circle, let Π: Δ×C→Δ be projection, and letA(Δ) be the algebra of complex functions continuous on Δ and analytic in int Δ. LetK be a compact set in C² such that Π(K)=Γ, and letK _λ≠{w∈C|(λ,w)∈K}. Suppose further that (a) for every λ∈Γ,K _λ is the union of two nonempty disjoint connected compact sets with connected complement, (b) there exists a function Q(λ,w)≠(w-R(λ))²-S(λ) quadratic in w withR,S∈A(Δ) such that for all λ∈Γ, {w∈C|Q(λ,w)=0}υ intK _λ, whereS has only one zero in int Δ, counting multiplicity, and (c) for every λ∈Γ, the map ω→Q(λ,ω) is injective on each component ofK _λ. Then we prove that К/K is the union of analytic disks 2-sheeted over int Δ, where К is the polynomial convex hull ofK. Furthermore, we show that БК/K is the disjoint union of such disks.     

Hausdorff operator of special kind on <Emphasis Type="Italic">p</Emphasis>-adic field and <Emphasis Type="Italic">BMO</Emphasis>-type spaces

For Hausdorff operator with generating function having support in the unit ball of p-adic field ℚ _p we give sufficient and necessary conditions of its boundedness in BMO-type spaces: BLO(ℚ _p ⁿ ), Q _r ^α,q (ℚ _p ⁿ ) and BMO _r ^α,q (ℚ _p ⁿ ). Some embedding relations between these spaces and Besov spaces are established.     

A hasse principle for function fields over pac fields

LetK be a perfect pseudo-algebraically closed field and letF be an extension ofK of relative transcendence degree 1. It is shown that the restriction map Res: Br(F)→Π_p Br(F _p ^h ) is injective, where p ranges over all non-trivialK-places ofF, andF _p ^h is the corresponding henselization. Conversely, the validity of this Hasse principle for all such extensionsF implies a weaker version of pseudo-algebraic closedness. As an application we determine the finitely generated pro-p closed subgroups of the absolute Galois group ofK(t).     

The smallest degree sum that yields potentially Kr,r-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(K_r,r,n) such that every n-term graphic sequence π = (d₁,d₂,...,d_n) with term sum σ(π) = d₁ + d₂ + ... + d_n ≥ σ(K_r,r,n) is potentially K_r,r-graphic, where K_r,r is an r × r complete bipartite graph, i.e. π has a realization G containing K_r,r as its subgraph. In this paper, the values σ(K_r,r,n) for even r and n ≥ 4r² - r - 6 and for odd r and n ≥ 4r² + 3r - 8 are determined.     

Rational points of bounded height on Del Pezzo surfaces of degree six

LetK be a number field. Denote byV ₃ a split Del Pezzo surface of degree six overK and by ω its canonical divisor. Denote byW ₃ the open complement of the exceptional lines inV ₃. LetN _{W s}(−ω, X) be the number ofK-rational points onW ₃ whose anticanonical heightH _−ω is bounded byX. Manin has conjectured that asymptoticallyN _{W 3}(−ω, X) tends tocX(logX)³, wherec is a constant depending only on the number field and on the normalization of the height. Our goal is to prove the following theorem: For each number fieldK there exists a constantc _K such thatN _{W 3}(−ω, X)≤c_KX(logX)^3+2r , wherer is the rank of the group of units ofO _K. The constantc _K is far from being optimal. However, ifK is a purely imaginary quadratic field, this proves an upper bound with a correct power of logX. The proof of Manin's conjecture for arbitrary number fields and a precise treatment of the constants would require a more sophisticated setting, like the one used by [Peyre] to prove Manin's conjecture and to compute the correct asymptotic constant (in some normalization) in the caseK=ℚ. Up to now the best result for arbitraryK goes back, as far as we know, to [Manin-Tschinkel], who gives an upper boundN _{W 3}(−ω,X)≤cX^l+ε. The author would like to express his gratitude to Daniel Coray and Per Salberger for their generous and indispensable support.     

Bounds on the number of cycles of length three in a planar graph

LetG be ap-vertex planar graph having a representation in the plane with nontriangular facesF ₁,F ₂, …,F _r. Letf ₁,f ₂, …,f _r denote the lengths of the cycles bounding the facesF ₁,F ₂, …,F _r respectively. LetC ₃(G) be the number of cycles of length three inG. We give bounds onC ₃(G) in terms ofp,f ₁,f ₂, …,f _r. WhenG is 3-connected these bounds are bounds for the number of triangles in a polyhedron. We also show that all possible values ofC ₃(G) between the maximum and minimum value are actually achieved. This research was supported in part by the U.S.A.F. Office of Scientific Research, Systems Command, under Grant AFOSR-76-3017 and the National Science Foundation under Grant ENG79-09724.     

On the topological types of symmetries of elliptic-hyperelliptic Riemann surfaces

LetX be a Riemann surface of genusg. The surfaceX is called elliptic-hyperelliptic if it admits a conformal involutionh such that the orbit spaceX/〈h〉 has genus one. The involutionh is then called an elliptic-hyperelliptic involution. Ifg>5 then the involutionh is unique, see [A]. We call symmetry to any anticonformal involution ofX. LetAut ^±(X) be the group of conformal and anticonformal automorphisms ofX and letσ, τ be two symmetries ofX with fixed points and such that {σ, hσ} and {τ, hτ} are not conjugate inAut ^±(X). We describe all the possible topological conjugacy classes of {σ, σh, τ, τh}. As consequence of our study we obtain that, in the moduli space of complex algebraic curves of genusg (g even >5), the subspace whose elements are the elliptic-hyperelliptic real algebraic curves is not connected. This fact contrasts with the result in [Se]: the subspace whose elements are the hyperelliptic real algebraic curves is connected. The authors are supported by BFM2002-04801.     

Cyclotomic and simplicial matroids

We show that two naturally occurring matroids representable over ℚ are equal: thecyclotomic matroid μ_n represented by then ^th roots of unity 1, ζ, ζ², …, ζ^n-1 inside the cyclotomic extension ℚ(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of ℚ-bases for ℚ(ζ) among then ^th roots of unity, which is tight if and only ifn has at most two odd prime factors. In addition, we study the Tutte polynomial of μ_n in the case thatn has two prime factors. First author supported by NSF Postdoctoral Fellowship. Second author supported by NSF grant DMS-0245379.     

The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces

Letf be aC ^r diffeomorphism,r≥2, of a two dimensional manifoldM ², and let Λ be a horseshoe off (i.e. a transitive and isolated hyperbolic set with topological dimension zero). We prove that there exist aC ^r neighborhoodU off and a neighbourhoodU of Λ such that forg∈U, the Hausdorff dimension of ∩ _n g ⁿ (U) is aC ^r−1 function ofg.     

Ergodic averages on circles

LetT ₁ andT ₂ be commuting invertible ergodic measure preserving flows on a probability space (X, A, μ). For t = (u,ν) ∈ ℝ², letT ^t =T ₁ ^u T ₂ ^v . LetS ¹ denote the unit circle in ℝ² and σ the rotation invariant unit measure on it. Then, forf∈Lp(X) withp>2, the averagesA _t f(x) = ∫ _s ¹ f(T ^ts x)σ(ds) conver the integral off for a. e.x, ast tends to 0 or infinity. This extends a result of R. Jones [J], who treated the case of three or more dimensions.