Pro-p Galois groups of rank ≤ 4

Let p be a prime > 2, let F be a field of characteristic ≠p containing a primitive p-th root of unity and let G _F (p) be the Galois group of the maximal Galois-p-extension of F. If rk G _F (p)≤ 4 then G _F (p) is a free pro-p product of metabelian groups or G _F (p) is a Demuškin group of rank 4. Received: 3 September 1997 / Revised version: 3 October 1997     

Pro-p galois groups of rank ≤4

Letp be a prime >2, letF be a field of characteristic ≠p containing a primitivep-th root of unity and letG _F (p) be the Galois group of the maximal Galois-p-extension ofF. Ifrk G _F (p)≤4 thenG _F (p) is a free pro-p product of metabelian groups orG _F (p) is a Demuškin group of rank 4.     

Separable subspaces of affine function spaces on convex compact sets

Let K be a compact convex subset of a separated locally convex space (over R) and let Ap(K) denote the space of all continuous real-valued affine mappings defined on K, endowed with the topology of pointwise convergence on the extreme points of K. In this paper we shall examine some topological properties of Ap(K). For example, we shall consider when Ap(K) is monolithic and when separable compact subsets of Ap(K) are metrizable.     

Morava K-Theories of p-Groups with Cyclic Maximal Subgroups and Other Related Groups

Let P be a non-Abelian finite p-group, p odd, with cyclic maximal subgroups, and let K(n)^*(–) denote the nth Morava K-theory at p. In this paper we determine the algebras K(n)^*(BP) and K(n)^*(BG) for all groups G with Sylow p-subgroups isomorphic to P, giving further evidence for the fact that Morava K-theory as an invariant of finite groups, is finer than ordinary modp cohomology. Mathematics Subject Classifications (2000): 55N20, 55N22.     

A discriminant criteria for reducibility of a polynomial

Letp(w) be a polynomial over a domainK. Ifp splits to linear factors then the discriminantD(p) is a square inK. In this paper we state an additional condition on the roots ofp which together with the discriminant condition imply the splitting ofp in case thatK=C [z] or Z. Some extensions are also discussed. Visiting Lady Davis Research Professor at the Hebrew University of Jerusalem, Fall Semester, 1984. Supported in part by US-Israel Binational Science Foundation grant 3225/84.     

Gröbner bases and triangulations of the second hypersimplex

The algebraic technique of Gröbner bases is applied to study triangulations of the second hypersimplex (2,n). We present a quadratic Gröbner basis for the associated toric idealK(K _n ). The simplices in the resulting triangulation of (2,n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding ofK _n . Forn6 the number of distinct initial ideals ofI(K _n ) exceeds the number of regular triangulations of (2,n); more precisely, the secondary polytope of (2,n) equals the state polytope ofI(K _n ) forn5 but not forn6. We also construct a non-regular triangulation of (2,n) forn9. We determine an explicit universal Gröbner basis ofI(K _n ) forn8. Potential applications in combinatorial optimization and random generation of graphs are indicated.Research partially supported by a doctoral fellowship of the National University of Mexico, the National Science Foundation, the David and Lucile Packard Foundation and the U.S. Army Research Office (through ACSyAM/MSI, Cornell).     

On Rubin’s variant of the <Emphasis Type="Italic">p</Emphasis>-adic Birch and Swinnerton–Dyer conjecture II

Let E/Q be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field K. In 1991, by studying a certain special value of the Katz two-variable p-adic L-function lying outside the range of p-adic interpolation, K. Rubin formulated a p-adic variant of the Birch and Swinnerton–Dyer conjecture when E(K) is infinite, and he proved that his conjecture is true for E(K) of rank one. When E(K) is finite, however, the statement of Rubin’s original conjecture no longer applies, and the relevant special value of the appropriate p-adic L-function is equal to zero. In this paper we extend our earlier work and give an unconditional proof of an analogue of Rubin’s conjecture when E(K) is finite.     

On Brown’s constant associated with irreducible polynomials over henselian valued fields

Let v be a henselian valuation of arbitrary rank of a field K and be the prolongation of v to the algebraic closure of K with value group . In 2008, Ron Brown gave a class P of monic irreducible polynomials over K such that to each g(x) belonging to P, there corresponds a smallest constant λ_g belonging to (referred to as Brown’s constant) with the property that whenever is more than λ_g with K(β) a tamely ramified extension of (K,v), then K(β) contains a root of g(x). In this paper, we determine explicitly this constant besides giving an important property of λ_g without assuming that K(β)/K is tamely ramified.     

A variant of the classical Ramsey problem

For fixed integersp, q an edge coloring of a complete graphK is called a (p, q)-coloring if the edges of everyK _p K are colored with at leastq distinct colors. Clearly, (p, 2)-colorings are the classical Ramsey colorings without monochromaticK _p subgraphs. Letf(n, p, q) be the minimum number of colors needed for a (p, q)-coloring ofK _n . We use the Local Lemma to give a general upper bound forf. We determine for everyp the smallestq for whichf(n, p, q) is linear inn and the smallestq for whichf(n, p, q) is quadratic inn. We show that certain special cases of the problem closely relate to Turán type hypergraph problems introduced by Brown, Erds and T. Sós. Other cases lead to problems concerning proper edge colorings of complete graphs.Supported by OTKA grant 16414.     

Restricted Elasticity and Rings of Integer-Valued Polynomials Determined by Finite Subsets

Let D be an integral domain such that Int(D) ≠ K[X] where K is the quotient field of D. There is no known example of such a D so that Int(D) has finite elasticity. If E is a finite nonempty subset of D, then it is known that Int(E, D) = {f(X) ∈ K[X] | f(e) ∈ D for all e ∈ E} is not atomic. In this note, we restrict the notion of elasticity so that it is applicable to nonatomic domains. For each real number r ≥ 1, we produce a ring of integer-valued polynomials with restricted elasticity r. We further show that if D is a unique factorization domain and E is finite with |E| > 1, then the restricted elasticity of Int(E, D) is infinite. Part of this work was completed while the first author was on an Academic Leave granted by the Trinity University Faculty Development Committee.     

Field theory for the function field of the quintic fermat curve

We study the structure of the function field K of the quintic Fermat curve F(5) in the following way: Let K m be a g-maximal rational subfield of K. Then the field extension K/K _m is obtained by the projection from F(5) to a line with a center p, ∈ F(5). By using this fact, we consider the field extension K/K m from several point of view.     

Integral Schur algebras forGL 2

The structure of Schur algebrasS(2,r) over the integral domainZ is intensively studied from the quasi-hereditary algebra point of view. We introduce certain new bases forS(2,r) and show that the Schur algebraS(2,r) modulo any ideal in the defining sequence is still such a Schur algebra of lower degree inr. A Wedderburn-Artin decomposition ofS _K (2,r) over a fieldK of characteristic 0 is described. Finally, we investigate the extension groups between two Weyl modules and classify the indecomposable Weyl-filtered modules for the Schur algebrasS _Zp(2,r) withr<p ² . Research supported by ARC Large Grant L20.24210     

Hyperelliptic jacobians with real multiplication

Let K be a field of characteristic p≠2, and let f(x) be a sextic polynomial irreducible over K with no repeated roots, whose Galois group is isomorphic to A₅. If the jacobian J(C) of the hyperelliptic curve C:y²=f(x) admits real multiplication over the ground field from an order of a real quadratic field D, then either its endomorphism algebra is isomorphic to D, or p>0 and J(C) is a supersingular abelian variety. The supersingular outcome cannot occur when p splits in D.     

Moving subvarieties by endomorphisms

Let K be an algebraically closed field of characteristic zero and let A be a semiabelian variety defined over K. Let End(A) be the ring of endomorphisms of A. Let X ⊂ A be a subvariety of smaller dimension. We show that does not equal A(K). In particular, we may take K to be countable.     

The Structure of the Closure of the Rational Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>(μ)

Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K°. Let μ be a positive finite measure with its support contained in K. For 1 ≤ q < ∞, let A^q(K, μ) denote the closure of A(K) in L^q(μ). The aim of this work is to study the structure of the space A^q(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for A^q(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for P^q(μ), the closure of polynomials in L^q(μ), as the special case when K is a closed disk containing the support of μ.     

On the α-amenability of hypergroups

Let UC(K) denote the Banach space of all bounded uniformly continuous functions on a hypergroup K. The main results of this article concern the α-amenability of UC(K) and quotient and product of hypergroups. It is also shown that a Sturm-Liouville hypergroup with a positive index is α-amenable if and only if α = 1. Author’s address: Dietlinden Stra?e 16, 80802 München, Germany     

Markov-type Inequalities for Surface Gradients of Multivariate Polynomials

Let K ^d be a compact set with a smooth boundary and consider a polynomial p of total degree n such that p_C(K)1. Then we show that D_Tp(x)=o(n²) for any x Bd K and T a tangential direction at x. Moreover, the o(n²) term is given in terms of the modulus of smoothness of Bd K.     

On the algebraic K-theory of the complex K-theory spectrum

Let p≥5 be a prime, let ku be the connective complex K-theory spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. In this paper we study the p-primary homotopy type of the spectrum K(ku) by computing its mod (p,v ₁) homotopy groups. We show that up to a finite summand, these groups form a finitely generated free module over the polynomial algebra \mathbbF_p[b]{\mathbb{F}}_{p}[b], where b is a class of degree 2p+2 defined as a “higher Bott element”.     

Group algebras and enveloping algebras with nonmatrix and semigroup identities

Let Kbe a field of characteristic p> 0. Denote by ω(R) the augmentation ideal of either a group algebra (R) = K[G] or a restricted enveloping algebra R= u(L) over K. We first characterize those Rfor which ω(R) satisfies a polynomial identity not satisfied by the algebra of all 2 × 2 matrices over K. Then, we examine those Rfor which U J(R) satisfies a semigroup identity (that is, a polynomial identity which can be written as the difference of two monomials).