Two Diophantine Birds with One Stone

Integer Solutions are found to the equations t²–3(a²,b², (a + b)², (a–b)²) = p², q², r², s². These lead surprisinglyto solutions to the equations u² + (c², d², (c + d)², (c –d)²) = p², q², v², w², with the same values of p and q.     

On the Connectedness of Self-Affine Tiles

Let T be a self-affine tile in Rⁿ defined by an integral expandingmatrix A and a digit set D. The paper gives a necessary andsufficient condition for the connectedness of T. The conditioncan be checked algebraically via the characteristic polynomialof A. Through the use of this, it is shown that in R², for anyintegral expanding matrix A, there exists a digit set D suchthat the corresponding tile T is connected. This answers a questionof Bandt and Gelbrich. Some partial results for the higher-dimensionalcases are also given.     

On a Parabolic Symmetry of Finite Coxeter Groups

Let (W, S be a finite Coxeter system, and let JS. Any wW hasa unique factorization w = w^J w_J, where w_j belongs to the parabolicsubgroup W_J generated by J, and w^J is of minimal length in thecoset wW_J. It is shown here that w_I = w^J if and only if w^I =w_J, for all I, J S. Furthermore, a similar symmetry propertyin arbitrary (W_I, W_J-double cosets is conjectured, which linksthis result to the Solomon descent algebra of W. 2000 MathematicsSubject Classification 20F55.     

Spines and Embeddings of n-Manifolds

Every compact, connected PL manifold Mⁿ, with MⁿØ, collapsesto a codimension-one subpolyhedron Q^n–1, called a spineof Mⁿ. The purpose of this paper is to prove that, if Q^n–1is appropriately chosen, one can reconstruct Mⁿ from Q^n–1,after taking the Cartesian product with an interval I=[0, 1].     

A Multiplicative Property of Quantum Flag Minors II

Let U⁺ be the plus part of the quantized enveloping algebraof a simple Lie algebra of type A_n and let B^* be the dual canonicalbasis of U⁺. Let b, b' be in B^*, and suppose that one of thetwo elements is a q-commuting product of quantum flag minors.It is shown that b and b' are multiplicative if and only ifthey q-commute.     

Local Rigidity of Infinite-Dimensional Teichmuller Spaces

This paper presents a rigidity theorem for infinite-dimensionalBergman spaces of hyperbolic Riemann surfaces, which statesthat the Bergman space A¹(M), for such a Riemann surface M,is isomorphic to the Banach space of summable sequence, l¹.This implies that whenever M and N are Riemann surfaces thatare not analytically finite, and in particular are not necessarilyhomeomorphic, then A¹(M) is isomorphic to A¹(N). It is knownfrom V. Markovic that if there is a linear isometry betweenA¹(M) and A¹(N), for two Riemann surfaces M and N of non-exceptionaltype, then this isometry is induced by a conformal mapping betweenM and N. As a corollary to this rigidity theorem presented here,taking the Banach duals of A¹(M) and l¹ shows that the spaceof holomorphic quadratic differentials on M, Q(M), is isomorphicto the Banach space of bounded sequences, l. As a consequenceof this theorem and the Bers embedding, the Teichmüllerspaces of such Riemann surfaces are locally bi-Lipschitz equivalent.     

A Criterion for Finite Topological Determinacy of Map-Germs

Let f, g: (Rⁿ, 0) (R^p, 0) be two C map-germs. Then f and gare C⁰-equivalent if there exist homeomorphism-germs h and lof (Rⁿ, 0) and (R^p, 0) respectively such that g = l f h^–1.Let k be a positive integer. A germ f is k-C⁰-determined ifevery germ g with j^k g(0) = j^k f(0) is C⁰-equivalent to f. Moreover,we say that f is finitely topologically determined if f is k-C⁰-determinedfor some finite k. We prove a theorem giving a sufficient conditionfor a germ to be finitely topologically determined. We explainthis condition below. Let N and P be two C manifolds. Consider the jet bundle J^k(N,P) with fiber J^k(n, p). Let z in J^k(n, p) and let f be suchthat z = j^kf(0). Define Whether (f) < k depends only on z, not on f. We can thereforedefine the set Let W^k(N, P) be the subbundle of J^k(N, P) with fiber W^k(n, p).Mather has constructed a finite Whitney (b)-regular stratificationS^k(n, p) of J^k(n, p) – W^k(n, p) such that all strata aresemialgebraic and K-invariant, having the property that if S^k(N,P) denotes the corresponding stratification of J^k(N, P) –W^k(N, P) and f C(N, P) is a C map such that j^kf is multitransverseto S^k(N, P), j^kf(N) W^k(N, P) = and N is compact (or f is proper),then f is topologically stable. For a map-germ f: (Rⁿ, 0) (R^p, 0), we define a certain ojasiewiczinequality. The inequality implies that there exists a representativef: U R^p such that j^kf(U – 0) W^k (Rⁿ, R^p = and suchthat j^kf is multitransverse to S^k (Rⁿ, R^p) at any finite setof points S U – 0. Moreover, the inequality controlsthe rate j^kf becomes non-transverse as we approach 0. We showthat if f satisfies this inequality, then f is finitely topologicallydetermined. 1991 Mathematics Subject Classification: 58C27.     

On Certain Exponential Sums and the Distribution of Diffie-Hellman Triples

Let g be a primitive root modulo a prime p. It is proved thatthe triples (g^x, g^y, g^xy), x, y = 1, ..., p–1, are uniformlydistributed modulo p in the sense of H. Weyl. This result isbased on the following upper bound for double exponential sums.Let >0 be fixed. Then uniformly for any integers a, b, c with gcd(a, b, c, p) = 1.Incomplete sums are estimated as well. The question is motivated by the assumption, often made in cryptography,that the triples (g^x, g^y, g^xy) cannot be distinguished fromtotally random triples in feasible computation time. The resultsimply that this is in any case true for a constant fractionof the most significant bits, and for a constant fraction ofthe least significant bits.     

Rank of Elliptic Curves over almost Separably Closed Fields

Let E be an elliptic curve over a finitely generated infinitefield K. Let K^s denote a separable closure of K, an elementof the Galois group G_K=Gal(K^s/K), and K^s() the invariant subfieldof K^s. If the characteristic of K is not 2 and belongs to asuitable open subgroup of G_K, then E(K^s()) has infinite rank.2000 Mathematics Subject Classification 11G05.     

On Artin's Conjecture, I: Systems of Diagonal Forms

As a special case of a well-known conjecture of Artin, it isexpected that a system of R additive forms of degree k, say [formula] with integer coefficients a_ij, has a non-trivial solution inQ_p for all primes p whenever [formula] Here we adopt the convention that a solution of (1) is non-trivialif not all the x_i are 0. To date, this has been verified onlywhen R=1, by Davenport and Lewis [4], and for odd k when R=2,by Davenport and Lewis [7]. For larger values of R, and in particularwhen k is even, more severe conditions on N are required toassure the existence of p-adic solutions of (1) for all primesp. In another important contribution, Davenport and Lewis [6]showed that the conditions [formula] are sufficient. There have been a number of refinements of theseresults. Schmidt [13] obtained N>>R²k³ log k, and Low,Pitman and Wolff [10] improved the work of Davenport and Lewisby showing the weaker constraints [formula] to be sufficient for p-adic solubility of (1). A noticeable feature of these results is that for even k, onealways encounters a factor k³ log k, in spite of the expectedk² in (2). In this paper we show that one can reach the expectedorder of magnitude k². 1991 Mathematics Subject Classification11D72, 11D79.     

Some Remarks on p-Adic Analytic Groups

A theorem of Maranda [1, Section 30] states that if F is a finitegroup, p is a prime and p^e exactly divides |F|, then a Z_pF-latticeM is determined up to isomorphism by its finite quotient M/p^e+1M.If M is a free Z_p-module of rank d, this is equivalent to sayingthat representations of F in GL_d(Z_p) are determined up to equivalenceby their images modulo p^e+1. 1991 Mathematics Subject Classification20E18, 22E20.     

Plane with A{infty}-Weighted Metric not Bilipschitz Embeddable to Rn

A planar set G R² is constructed that is bilipschitz equivalentto (G,d^z), where (G, d) is not bilipschitz embeddable to anyuniformly convex Banach space. Here, Z (0, 1) and d^z denotesthe zth power of the metric d. This proves the existence ofa strong A weight in R², such that the corresponding deformedgeometry admits no bilipschitz mappings to any uniformly convexBanach space. Such a weight cannot be comparable to the Jacobianof a quasiconformal self-mapping of R². 2000 Mathematics SubjectClassification 54E40 (primary); 30C62, 30C65, 28A80 (secondary).     

Squared Givens Rotation

Given a matrix A, let U be the upper triangular matrix thatresults from the application of Gaussian elimination to A^TA.We develop a fast and square-root-free Givens rotation admittingthe direct calculation of U from A.     

Rank Distributions on Coarse Spaces and Ideal Structure of Roe Algebras

The notions of controlled truncations for operators in the Roealgebras C^* (X) of a coarse space (X, ) with uniformly locallyfinite coarse structure, and rank distributions on (X, ) areintroduced. It is shown that the controlled propagation operatorsin an ideal I of C^* (X) are exactly the controlled truncationsof elements in I. It follows that the lattice of the idealsof C^* (X) in which controlled propagation operators are denseis isomorphic to the lattice of all rank distributions on (X,). If X is a discrete metric space with Yu's property A, thenthe ideal structure of the Roe algebra C^* (X is completely determinedby the rank distributions on X. 2000 Mathematics Subject Classification46L80, 46L89.     

Multiquadratic Extensions, Rigid Fields and Pythagorean Fields

Let F be a field of characteristic other than 2. Let F⁽²⁾ denotethe compositum over F of all quadratic extensions of F, letF⁽³⁾ denote the compositum over F⁽²⁾ of all quadratic extensionsof F⁽²⁾ that are Galois over F, and let F^{3} denote the compositumover F⁽²⁾ of all quadratic extensions of F⁽²⁾. This paper showsthat F⁽³⁾ = F^{3} if and only if F is a rigid field, and thatF⁽³⁾ = K⁽³⁾ for some extension K of F if and only if F is Pythagoreanand . The proofs depend mainly on the behavior of quadratic forms over quadratic extensions,and the corresponding norm maps.     

Canonical curves in P3

Let Hilb^6t–3(P³) be the Hilbert scheme of closed 1-dimensionalsubschemes of degree 6 and arithmetic genus 4 in P³. Let H bethe component of Hilb^6t–3(P³) whose generic point correspondsto a canonical curve, that is, a complete intersection of aquadric and a cubic surface in P³. Let F be the vector spaceof linear forms in the variables z₁, z₂, z₃, z₄. Denote by F_dthe vector space of homogeneous forms of degree d. Set X = (f₂,f₃)where f₂ P(F₂) is a quadric surface, and f₃ P(F₃/f₂ ·F) is a cubic modulo f₂. Wehave a rational map, : X ... Hdefined by (f₂,f₃) f₂ f₃. It fails to be regular along thelocus where f₂ and f₃ acquire a common linear component. Ourmain result gives an explicit resolution of the indeterminaciesof as well as of the singularities of H. 2000 Mathematical Subject Classification: 14C05, 14N05, 14N10,14N15.     

The Uniqueness of the Prime Markoff Numbers

Given the Diophantine equation a²+b²+c²=3abc, a solution tripleof natural numbers (a, b, c) can be arranged in ascending orderso that abc. Then, given the largest element c, one can askwhether this uniquely determines the triple. This is referredto as the Markoff conjecture. The paper proves that, if c isprime, then there is indeed only one triple that solves theequation with c as the largest element. The proof uses onlystandard algebraic number theory, but it was prompted by geometricconsiderations.     

The Decomposition of Lie Powers

Let G be a group, F a field of prime characteristic p and Va finite-dimensional FG-module. Let L(V) denote the free Liealgebra on V regarded as an FG-submodule of the free associativealgebra (or tensor algebra) T(V). For each positive integerr, let L^r (V) and T^r (V) be the rth homogeneous components ofL(V) and T(V), respectively. Here L^r (V) is called the rth Liepower of V. Our main result is that there are submodules B₁,B₂, ... of L(V) such that, for all r, B_r is a direct summandof T^r(V) and, whenever m 0 and k is not divisible by p, themodule is the direct sum of , . Thus every Lie power is a direct sum of Lie powers of p-powerdegree. The approach builds on an analysis of T^r (V) as a bimodulefor G and the Solomon descent algebra. 2000 Mathematics SubjectClassification 17B01 (primary), 20C07, 20C20 (secondary).     

Boundary Accessibility of a Domain Quasiconformally Equivalent to a Ball

A boundary point of a domain D in Rⁿ is said to be broadly accessibleif it ‘almost lies’ on the boundary of a round ballcontained in D. If f is a quasiconformal mapping of the unitball Bⁿ onto D, then it is shown that broadly accessible boundarypoints on D correspond under f to a set of full measure on Bⁿ.2000 Mathematics Subject Classification 30C65.     

Primes Represented by Binary Cubic Forms

Let f(x, y) be a binary cubic form with integral rational coefficients,and suppose that the polynomial f(x, y) is irreducible in Q[x,y] and no prime divides all the coefficients of f. We provethat the set f Z⁽²⁾ contains infinitely many primes unless f(a,b) is even for each (a,b) in Z², in which case the set contains infinitely many primes. 2000Mathematical Subject Classification: primary 11N32; secondary11N36, 11R44.