Rational Points on a Class of Superelliptic Curves

A famous Diophantine equation is given by y^k=(x+1)(x+2)...(x+m). (1) For integers k2 and m2, this equation only has the solutionsx = –j (j = 1, ..., m), y = 0 by a remarkable result ofErds and Selfridge [9] in 1975. This put an end to the old questionof whether the product of consecutive positive integers couldever be a perfect power (except for the obviously trivial cases).In a letter to D. Bernoulli in 1724, Goldbach (see [7, p. 679])showed that (1) has no solution with x0 in the case k = 2 andm = 3. In 1857, Liouville [18] derived from Bertrand's postulatethat for general k2 and m2, there is no solution with x0 ifone of the factors on the right-hand side of (1) is prime. Byuse of the Thue–Siegel theorem, Erds and Siegel [10] provedin 1940 that (1) has only trivial solutions for all sufficientlylarge kk₀ and all m. This was closely related to Siegel's earlierresult [30] from 1929 that the superelliptic equation y^k=f(x) has at most finitely many integer solutions x, y under appropriateconditions on the polynomial f(x). The ineffectiveness of k₀was overcome by Baker's method [1] in 1969 (see also [2]). In 1955, Erds [8] managed to re-prove the result jointly obtainedwith Siegel by elementary methods. A refinement of Erds' ideasfinally led to the above-mentioned theorem as follows.     

Oscillation Properties of a Semi-discrete Approximation to the Beam Equation with a Second Order Term

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx–[p(x)u_x]_x=0, 0< x < L, t > 0, where it is assumed that w, p,and q are positive on the interval [0, L], is approximated bythe method of straight lines. The resulting approximation isa linear system of differential equations with coefficient matrixS. The matrix S is studied under a variety of boundary conditionswhich result in a conservative system. In all cases the matrixS is shown to be similar to an oscillation matrix.     

Structural Stability of Constrained Polynomial Systems

The structural stability of constrained polynomial differentialsystems of the form a(x, y)x'+b(x, y)y'=f(x, y), c(x, y)x'+d(x,y)y'=g(x, y), under small perturbations of the coefficientsof the polynomial functions a, b, c, d, f and g is studied.These systems differ from ordinary differential equations at‘impasse points’ defined by ad–bc=0. Extensionsto this case of results for smooth constrained differentialsystems [7] and for ordinary polynomial differential systems[5] are achieved here. 1991 Mathematics Subject Classification34C35, 34D30.     

Irregularities of Point Distribution Relative to Convex Polygons III

Suppose that P is a distribution of N points in the unit squareU=[0, 1]². For every x=(x₁, x₂)U, let B(x)=[0, x₁]x[0, x₂] denotethe aligned rectangle containing all points y=(y₁, y₂)U satisfying0y₁x₁ and 0y₂x₂. Denote by Z[P; B(x)] the number of points ofP that lie in B(x), and consider the discrepancy function D[P; B(x)]=Z[P; B(x)]–Nµ(B(x)), where µ denotes the usual area measure.     

Hypersurface Singularities with 2-Dimensional Critical Locus

Consider an analytic germ f:(C^m, 0)(C, 0) (m3) whose criticallocus is a 2-dimensional complete intersection with an isolatedsingularity (icis). We prove that the homotopy type of the Milnorfiber of f is a bouquet of spheres, provided that the extendedcodimension of the germ f is finite. This result generalizesthe cases when the dimension of the critical locus is zero [8],respectively one [12]. Notice that if the critical locus isnot an icis, then the Milnor fiber, in general, is not homotopicallyequivalent to a wedge of spheres. For example, the Milnor fiberof the germ f:(C⁴, 0)(C, 0), defined by f(x₁, x₂, x₃, x₄) =x₁x₂x₃x₄ has the homotopy type of S¹xS¹xS¹. On the other hand,the finiteness of the extended codimension seems to be the rightgeneralization of the isolated singularity condition; see forexample [9–12, 17, 18]. In the last few years different types of ‘bouquet theorems’have appeared. Some of them deal with germs f:(X, x)(C, 0) wheref defines an isolated singularity. In some cases, similarlyto the Milnor case [8], F has the homotopy type of a bouquetof (dim X–1)-spheres, for example when X is an icis [2],or X is a complete intersection [5]. Moreover, in [13] Siersmaproved that F has a bouquet decomposition FF₀Sⁿ...Sⁿ (whereF₀ is the complex link of (X, x)), provided that both (X, x)and f have an isolated singularity. Actually, Siersma conjecturedand Tibr proved [16] a more general bouquet theorem for thecase when (X, x) is a stratified space and f defines an isolatedsingularity (in the sense of the stratified spaces). In thiscase F_iF_i, where the F_i are repeated suspensions of complexlinks of strata of X. (If (X, x) has the ‘Milnor property’,then the result has been proved by Lê; for details see[6].) In our situation, the space-germ (X, x) is smooth, but f hasbig singular locus. Surprisingly, for dim Sing f^–1(0)2,the Milnor fiber is again a bouquet (actually, a bouquet ofspheres, maybe of different dimensions). This result is in thespirit of Siersma's paper [12], where dim Sing f^–1(0)= 1. In that case, there is only a rather small topologicalobstruction for the Milnor fiber to be homotopically equivalentto a bouquet of spheres (as explained in Corollary 2.4). Inthe present paper, we attack the dim Sing f^–1(0) = 2 case.In our investigation some results of Zaharia are crucial [17,18].     

The Pontryagin Rings of Moduli Spaces of Arbitrary Rank Holomorphic Bundles Over a Riemann Surface

The cohomology of M(n, d), the moduli space of stable holomorphicbundles of coprime rank n and degree d and fixed determinant,over a Riemann surface of genus g 2, has been widely studiedfrom a wide range of approaches. Narasimhan and Seshadri [17]originally showed that the topology of M(n, d) depends onlyon the genus g rather than the complex structure of . An inductivemethod to determine the Betti numbers of M(n, d) was first givenby Harder and Narasimhan [7] and subsequently by Atiyah andBott [1]. The integral cohomology of M(n, d) is known to haveno torsion [1] and a set of generators was found by Newstead[19] for n = 2, and by Atiyah and Bott [1] for arbitrary n.Much progress has been made recently in determining the relationsthat hold amongst these generators, particularly in the ranktwo, odd degree case which is now largely understood. A setof relations due to Mumford in the rational cohomology ringof M(2, 1) is now known to be complete [14]; recently severalauthors have found a minimal complete set of relations for the‘invariant’ subring of the rational cohomology ofM(2, 1) [2, 13, 20, 25]. Unless otherwise stated all cohomology in this paper will haverational coefficients.     

A Geometric Invariant for Metabelian Pro-p Groups

In [2] Bieri and Strebel introduced a geometric invariant forfinitely generated abstract metabelian groups that determineswhich groups are finitely presented. For a valuable survey oftheir results, see [6]; we recall the definition briefly inSection 4. We shall introduce a similar invariant for pro-pgroups. Let F be the algebraic closure of F_p and U be the formal powerseries algebra F[T], with group of units U^x. Let Q be a finitelygenerated abelian pro-p group. We write Z_p[Q] for the completedgroup algebra of Q over Z_p. Let T(Q) be the abelian group Hom(Q,U^x) of continuous homomorphisms from Q to U^x. We write 1 forthe trivial homomorphism. Each vT(Q) extends to a unique continuousalgebra homomorphism from Z_p[Q]to U.     

On Simultaneous Diagonal Inequalities

Let F₁, ..., F_t be diagonal forms of degree k with real coefficientsin s variables, and let be a positive real number. The solubilityof the system of inequalities |F₁(x)|<,...,|F_t(x)|< in integers x₁, ..., x_s has been considered by a number of authorsover the last quarter-century, starting with the work of Cook[9] and Pitman [13] on the case t = 2. More recently, Brüdernand Cook [8] have shown that the above system is soluble providedthat s is sufficiently large in terms of k and t and that theforms F₁, ..., F_t satisfy certain additional conditions. Whathas not yet been considered is the possibility of allowing theforms F₁, ..., F_t to have different degrees. However, with therecent work of Wooley [18,20] on the corresponding problem forequations, the study of such systems has become a feasible prospect.In this paper we take a first step in that direction by studyingthe analogue of the system considered in [18] and [20]. Let₁, ..., _s and µ₁, ..., µ_s be real numbers such thatfor each i either _i or µ_i is nonzero. We define the forms and consider the solubility of the system of inequalities in rational integers x₁, ..., x_s. Although the methods developedby Wooley [19] hold some promise for studying more general systems,we do not pursue this in the present paper. We devote most ofour effort to proving the following theorem.     

One Cubic Diophantine Inequality

Suppose that G(x) is a form, or homogeneous polynomial, of odddegree d in s variables, with real coefficients. Schmidt [15]has shown that there exists a positive integer s₀(d), whichdepends only on the degree d, so that if s s₀(d), then thereis an x Z^s\{0} satisfying the inequality |G(x)|<1. (1) In other words, if there are enough variables, in terms of thedegree only, then there is a nontrivial solution to (1). Lets₀(d) be the minimum integer with the above property. In thecourse of proving this important result, Schmidt did not explicitlygive upper bounds for s₀(d). His methods do indicate how todo so, although not very efficiently. However, in fact muchearlier, Pitman [13] provided explicit bounds in the case whenG is a cubic. We consider a general cubic form F(x) with realcoefficients, in s variables, and look at the inequality |F(x)|<1. (2) Specifically, Pitman showed that if s(1314)²⁵⁶–1, (3) then inequality (2) is non-trivially soluble in integers. Wepresent the following improvement of this bound.     

Residually Finite Groups with all Subgroups Subnormal

The following result is established. THEOREM. Let G be a periodic, residually finite group with allsubgroups sub-normal. Then G is nilpotent. The well-known groups of Heineken and Mohamed [1] show thatthe hypothesis of residual finiteness cannot be omitted here,while examples in [5] show that a residually finite group withall subgroups subnormal need not be nilpotent. The proof ofthe Theorem will use the results of Möhres that a groupwith all subgroups subnormal is soluble [3] and that a periodichypercentral group with all subgroups subnormal is nilpotent[4]. Borrowing an idea from [2], the plan is to construct certainsubgroups H and K that intersect trivially, and to show thatthe subnormality of both leads to a contradiction. 1991 MathematicsSubject Classification 20E15.     

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.     

A Necessary and Sufficient Condition for a Linear Differential System to be Strongly Monotone

In order to present the results of this note, we begin withsome definitions. Consider a differential system [formula] where IR is an open interval, and f(t, x), (t, x)IxRⁿ, is acontinuous vector function with continuous first derivativesf_r/x_s, r, s=1, 2, ..., n. Let D_xf(t, x), (t, x)IxRⁿ, denote the Jacobi matrix of f(t,x), with respect to the variables x₁, ..., x_n. Let x(t, t₀,x₀), tI(t₀, x₀) denote the maximal solution of the system (1)through the point (t₀, x₀)IxRⁿ. For two vectors x, yRⁿ, we use the notations x>y and x>>yaccording to the following definitions: [formula] An nxn matrix A=(a_rs) is called reducible if n2 and there existsa partition [formula] (p1, q1, p+q=n) such that [formula] The matrix A is called irreducible if n=1, or if n2 and A isnot reducible. The system (1) is called strongly monotone if for any t₀I, x₁,x₂Rⁿ [formula] holds for all t>t₀ as long as both solutions x(t, t₀, x_i),i=1, 2, are defined. The system is called cooperative if forall (t, x)IxRⁿ the off-diagonal elements of the nxn matrix D_xf(t,x) are nonnegative. 1991 Mathematics Subject Classification34A30, 34C99.     

Weak Cayley Tables

In [1] Brauer puts forward a series of questions on group representationtheory in order to point out areas which were not well understood.One of these, which we denote by (B1), is the following: whatinformation in addition to the character table determines a(finite) group? In previous papers [5, 7–13], the originalwork of Frobenius on group characters has been re-examined andhas shed light on some of Brauer's questions, in particularan answer to (B1) has been given as follows. Frobenius defined for each character of a group G functions^(k):G^(k) C for k = 1, ..., deg with ⁽¹⁾ = . These functionsare called the k-characters (see [10] or [11] for their definition).The 1-, 2- and 3-characters of the irreducible representationsdetermine a group [7, 8] but the 1- and 2-characters do not[12]. Summaries of this work are given in [11] and [13].     

One-Dimensional p-Adic Subanalytic Sets

In this paper we extend two theorems from [2] on p-adic subanalyticsets, where p is a fixed prime number, Q_p is the field of p-adicnumbers and Z_p is the ring of p-adic integers. One of thesetheorems [2, 3.32] says that each subanalytic subset of Z_p issemialgebraic. This is extended here as follows.     

An Optimal Representation Formula for Carnot-Caratheodory Vector Fields

The simplest example of the sort of representation formula thatwe shall study is the following familiar inequality for a smooth,real-valued function f(x) defined on a ball B in N-dimensionalEuclidean space R^N: [formula] where f denotes the gradient of f, f_B is the average |B|^–1_Bf(y)dy, |B| is the Lebesgue measure of B, and C is a constantwhich is independent of f, x and B. This formula can be found,for example, in [4] and [12]; see also the closely related estimatesin [20, pp. 228{231]. Indeed, such a formula holds in any boundedconvex domain. 1991 Mathematics Subject Classification 31B10,46E35, 35A22.     

An Euler-Type Version of the Local Steiner Formula for Convex Bodies

The purpose of this note is to establish a new version of thelocal Steiner formula and to give an application to convex bodiesof constant width. This variant of the Steiner formula generalizesresults of Hann [3] and Hug [6], who use much less elementarytechniques than the methods of this paper. In fact, Hann askedfor a simpler proof of these results [4, Problem 2, p. 900].We remark that our formula can be considered as a Euclideananalogue of a spherical result proved in [2, p. 46], and thatour method can also be applied in hyperbolic space. For some remarks on related formulas in certain two-dimensionalMinkowski spaces, see Hann [5, p. 363]. For further information about the notions used below, we referto Schneider's book [9]. Let Kⁿ be the set of all convex bodiesin Euclidean space Rⁿ, that is, the set of all compact, convex,non-empty subsets of Rⁿ. Let S^n–1 be the unit sphere.For KKⁿ, let NorK be the set of all support elements of K, thatis, the pairs (x, u)RⁿxS^n–1 such that x is a boundarypoint of K and u is an outer unit normal vector of K at thepoint x. The support measures (or generalized curvature measures)of K, denoted by ₀(K.), ..., _n–1(K.), are the unique Borelmeasures on RⁿxS^n–1 that are concentrated on NorK andsatisfy [formula] for all integrable functions f:RⁿR; here denotes the Lebesguemeasure on Rⁿ. Equation (1), which is a consequence and a slightgeneralization of Theorem 4.2.1 in Schneider [9], is calledthe local Steiner formula. Our main result is the following.1991 Mathematics Subject Classification 52A20, 52A38, 52A55.     

Corrigendum

SOLUTIONS OF p-SUBLINEAR p-LAPLACIAN EQUATION VIA MORSE THEORY (J. London Math. Soc. (2) 72 (2005) 632–644) YUXIA GUO AND JIAQUAN LIU The above-mentioned paper was influenced by the work of V. Moroz[1] and Z. Q. Wang [2], which had been inadvertently omittedfrom the bibliography.     

Limits Along Parallel Lines and the Classical Fine Topology

The fine topology on Rⁿ (n2) is the coarsest topology for whichall superharmonic functions on Rⁿ are continuous. We refer toDoob [11, 1.XI] for its basic properties and its relationshipto the notion of thinness. This paper presents several theoremsrelating the fine topology to limits of functions along parallellines. (Results of this nature for the minimal fine topologyhave been given by Doob – see [10, Theorem 3.1] or [11,1.XII.23] – and the second author [15].) In particular,we will establish improvements and generalizations of resultsof Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihland Seidel [6], Schneider [21], Berman [7], and Armitage andNelson [4], and will also solve a problem posed by the latterauthors. An early version of our first result is due to Evans [12, p.234], who proved that, if u is a superharmonic function on R³,then there is a set ER²x{0}, of two-dimensional measure 0, suchthat u(x, y,·) is continuous on R whenever (x, y, 0)E.We denote a typical point of Rⁿ by X=(X' x), where X'R^n–1and xR. Let :RⁿR^n–1x{0} denote the projection map givenby (X', x) = (X', 0). For any function f:Rⁿ[–, +] andpoint X we define the vertical and fine cluster sets of f atX respectively by C_V(f;X)={l[–, +]: there is a sequence (t_m) of numbersin R\{x} such that t_mx and f(X', t_m)l}| and C_F(f;X)={l[–, +]: for each neighbourhood N of l in [–,+], the set f^–1(N) is non-thin at X}. Sets which are open in the fine topology will be called finelyopen, and functions which are continuous with respect to thefine topology will be called finely continuous. Corollary 1(ii)below is an improvement of Evans' result.     

Corrigenda: Mackey Functors And Control of Fusion

In the introduction to [3] it is mistakenly claimed that Mislin'sproof uses Carlsson's proof of Segal's Burnside ring conjecture.In fact, Mislin uses instead the work of Carlsson, Miller andLannes on Sullivan's fixed-point conjecture, and work of Dwyerand Zabrodski [1]. There is, however, a proof by Snaith [2]that depends on a version of the Segal conjecture. 2000 MathematicsSubject Classification 20J06 (primary), 20C05 (secondary).     

Derivations on Real and Complex JB*-Triples

At the regional conference held at the University of California,Irvine, in 1985 [24], Harald Upmeier posed three basic questionsregarding derivations on JB*-triples: (1) Are derivations automatically bounded? (2) When are all bounded derivations inner? (3) Can bounded derivations be approximated by inner derivations? These three questions had all been answered in the binary cases.Question 1 was answered affirmatively by Sakai [17] for C*-algebrasand by Upmeier [23] for JB-algebras. Question 2 was answeredby Sakai [18] and Kadison [12] for von Neumann algebras andby Upmeier [23] for JW-algebras. Question 3 was answered byUpmeier [23] for JB-algebras, and it follows trivially fromthe Kadison–Sakai answer to question 2 in the case ofC*-algebras. In the ternary case, both question 1 and question 3 were answeredby Barton and Friedman in [3] for complex JB*-triples. In thispaper, we consider question 2 for real and complex JBW*-triplesand question 1 and question 3 for real JB*-triples. A real orcomplex JB*-triple is said to have the inner derivation propertyif every derivation on it is inner. By pure algebra, every finite-dimensionalJB*-triple has the inner derivation property. Our main results,Theorems 2, 3 and 4 and Corollaries 2 and 3 determine whichof the infinite-dimensional real or complex Cartan factors havethe inner derivation property.