Additive representations of natural numbers by mixtures of squares, cubes and biquadrates belong to the class of more interesting special cases which form the object of attention for testing the general expectation that any sufficiently large natural number n is representable in the form

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as soon as the reciprocal sum [sum ]sj=1k−1j is reasonably large. With the exception of a handful of very special problems, in the current state of knowledge the latter reciprocal sum must exceed 2, at the very least, in order that it be feasible to successfully apply the Hardy–Littlewood method to treat the corresponding additive problem. Here we remove a case from the list of those combinations of exponents which have defied treatment thus far.

Let K be a number field with ring of integers R. For each integer g>1 we consider the collection of abelian, étale R-coverings f[ratio ]Y→X, where X and Y are connected proper curves over R and the genus of X is g. We ask the following question: is there a positive integer B = B(K, g) which bounds the degree of such coverings? In this note we provide partial results towards such a bound and study the relationship with bounds on torsion in abelian varieties.

We study the Diophantine equation

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in integers x, y > 1, n > 2 and q [ges ] 2. Without loss of generality, we suppose that q is prime.

Saradha and Shorey [SS] considered equation (1·1) with x = z2 and proved that it has no solutions in the cases z > 31 or z ∈ {2, 3, 4, 8, 9, 16, 27}.Notice that the proof for z ∈ {2, 4, 8, 16} is elementary, whereas for z ∈ {3, 9, 27} the proof uses a result of Darmon–Merel [DM] which is a generalization of Wiles' result on Fermat's theorem. The purpose of the present paper is to treat completely the remaining cases and to prove the following result.

A Noetherian integral domain R is said to be a splinter if it is a direct summand, as an R-module, of every module-finite extension ring (see [Ma]). In the case that R contains the field of rational numbers, it is easily seen that R is splinter if and only if it is a normal ring, but the notion is more subtle for rings of characteristic p>0. It is known that F-regular rings of characteristic p are splinters and Hochster and Huneke showed that the converse is true for locally excellent Gorenstein rings [HH4]. In this paper we extend their result by showing that ℚ-Gorenstein splinters are F-regular. Our main theorem is:

THEOREM 1.1. Let R be a locally excellent ℚ-Gorenstein integral domain of characteristic p>0. Then R is F-regular if and only if it is a splinter.

A longstanding conjecture claims that the Diophantine equation

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has finitely many solutions and, maybe, only those given by

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Among the known results, let us mention that Ljunggren [9] solved (1) completely when q = 2 and that very recently Bugeaud et al. [3] showed that (1) has no solution when x is a square. For more information and in particular for finiteness type results under some extra hypotheses, we refer the reader to Nagell [11], Shorey & Tijdeman [16, 17] and to the recent survey of Shorey [15]. One of the main tools used in most of the proofs is Baker's theory of linear form in archimedean logarithms of algebraic numbers and especially a dramatic sharpening obtained when the algebraic numbers involved are closed to 1. This was first noticed by Shorey [14], and has been applied in numerous works relating to (1) or to the Diophantine equation

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see for instance [1, 5, 10]. The purpose of the present work is to prove a similar sharpening for linear forms in p-adic logarithms and to show how it can be applied in the context of (1). Further, following previous investigations by Sander [12] and Saradha & Shorey [13], we derive from our results an irrationality statement for Mahler's numbers.

In [R2] we showed how elementary considerations involving geometry on ruled surfaces may be used to obtain recursive enumerative formulae for rational plane curves. Here we show how similar considerations may be used to obtain further enumerative formulae, as follow. First some notation. As usual we denote by Ngd the number of irreducible plane curves of degree d and genus g through 3d+g−1 general points. Also, we denote by Ngd→ (resp. Ngd×) the number of such curves passing through general points A1, …, A3d+g−2 and having a given tangent direction (resp. a node) at A1. As is well known and easy to see, we have

formula here

For any d, g, these numbers are computed in [R1] as part of a more general recursive procedure. For N0d, N1d, relatively simple recursions have been given by Kontsevich–Manin (see [FP]) and Eguchi–Hori–Xiong–Getzler (see [P1]), respectively.

Flips occur in the theory of minimal models of algebraic varieties. For an introduction and references see [1, lecture no. 5]. For varieties X− and X+, I denote the canonical class by K− and respectively. A flip is a diagram X−→X←X+ of normal complex quasiprojective 3-folds satisfying the conditions:

1. both morphisms are birational and projective, contracting only finitely many curves C±⊂X± to an isolated singular point P∈X;

2. the divisors −K− and K+ are relatively ample, that is, −K−Γ>0 for any curve Γ contracted by the morphism X−→X and similarly for K+;

3. the two varieties X− and X+ have only terminal singularities.

A diagram satisfying condition 1 is called a flip diagram. It is said to be directed by the canonical class if it also satisfies condition 2. Notice that condition 3 is overstated since under all the other conditions X+ will automatically have terminal singularities (see [4, 5-1-11(2)]).

We give a characterization in terms of finitely many arithmetical invariants for two finitely generated, torsion-free, nilpotent groups of nilpotency class two and Hirsch length six to have isomorphic localizations at every finite set of primes. This result is obtained through adequate reduction of an arbitrary binary integral quadratic form. We derive some consequences and, in particular, we characterize completely those groups N in the above class of groups, for which the Mislin genus coincides with the strong genus (i.e. those groups N for which, whenever a nilpotent group M satisfies Mp ≅ Np for every prime p, then MP ≅ NP for every finite set of primes P).

Let the elliptic curve E be defined by the equation

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with a1, …, a6 ∈ ℤ. Define a finite set of places S = {q1, …, qs−1, qs = ∞} of ℚ and put Q = max {q1, …, qs−1}. Let E(ℚ) denote the set of (x, y) ∈ ℚ2 satisfying (1) and the infinite point [Oscr ].

The multiplicative height of a rational point P = (x, y) ∈ E(ℚ) is defined as the following product over all places q of ℚ (including q = ∞):

formula here

where the [mid ]x[mid ]qs are the normalized multiplicative absolute values of ℚ corresponding to the places q.

Given a finitely presented group G and an epimorphism χ:G→Z, constraints on the orders of automorphisms F:G→G such that χ∘F = χ are obtained via symbolic dynamics. The techniques provide new obstructions to periodicity for knots and links.

The kth Fitting ideal of the Alexander invariant B of an arrangement [Ascr ] of n complex hyperplanes defines a characteristic subvariety, Vk([Ascr ]), of the algebraic torus ([Copf ]*)n. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of Vk([Ascr ]). For any arrangement [Ascr ], we show that the tangent cone at the identity of this variety coincides with [Rscr ]1k(A), one of the cohomology support loci of the Orlik–Solomon algebra. Using work of Arapura [1], we conclude that all irreducible components of Vk([Ascr ]) which pass through the identity element of ([Copf ]*)n are combinatorially determined, and that [Rscr ]1k(A) is the union of a subspace arrangement in [Copf ]n, thereby resolving a conjecture of Falk [11]. We use these results to study the reflection arrangements associated to monomial groups.

The main result of this paper establishes invariance under change of generators for automatic structures for monoids (a property that is well known to hold for automatic groups but fails for semigroups). This result is then applied to show that if a free product of two monoids is automatic, then so are both the free factors. Finally, the difference between automatic structures, in terms of monoid generating sets and semigroup generating sets, is discussed.

The main result of [15] is that the classifying space of the stable mapping class group after plus construction BΓ+∞ is an infinite loop space. This result is used to show that, localized away from two, a connected component of the stable homotopy groups of spheres QS0 splits off BΓ+∞. The splitting is a splitting of infinite loop spaces. It follows immediately that the homology with coefficients in ℤ[½] of the infinite symmetric group is a direct summand of the homology of the stable mapping class group.

In the mid 1970s Mark Mahowald constructed a new infinite family of elements in the 2-component of the stable homotopy groups of spheres, ηj∈πSj2 (S0)(2) [M]. Using standard Adams spectral sequence terminology (which will be recalled in Section 3 below), ηj is detected by h1hj∈Ext2,*[Ascr ] (Z/2, Z/2). Thus he had found an infinite family of elements all having the same Adams filtration (in this case, 2), thus dooming the so-called Doomsday Conjecture. His constructions were ingenious: his elements were constructed as composites of pairs of maps, with the intermediate spaces having, on one hand, a geometric origin coming from double loopspace theory and, on the other hand, mod2 cohomology making them amenable to Adams Spectral Sequence analysis and suggesting that they were related to the new discovered Brown–Gitler spectra [BG].

In the years that followed, various other related 2-primary infinite families were constructed, perhaps most notably (and correctly) Bruner's family detected by h2h2j∈ Ext3,*[Ascr ](Z/2, Z/2) [B]. An odd prime version was studied by Cohen [C], leading to a family in πS∗(S0)(p) detected by h0bj∈ Ext3,*[Ascr ] (Z/p, Z/p) and a filtration 2 family in the stable homotopy groups of the odd prime Moore space. Cohen also initiated the development of odd primary Brown–Gitler spectra, completed in the mid 1980s, using a different approach, by Goerss [G], and given the ultimate ‘modern’ treatment by Goerss, Lannes and Morel in the 1993 paper [GLM]. Various papers in the late 1970s and early 1980s, e.g. [BP, C, BC], related some of these to loopspace constructions.

Our project originated with two goals. One was to see if any of the later work on Brown–Gitler spectra led to clarification of the original constructions. The other was to see if taking advantage of post Segal Conjecture knowledge of the stable cohomotopy of the classifying space BZ/p would help in constructing new families at odd primes, in particular a conjectural family detected by h0hj∈ Ext2,*[Ascr ] (Z/p, Z/p). (This followed a paper [K1] by one of us on 2 primary families from this point of view.)

For each positive integer n, let Fn be a free group of rank n with basis (in other words, free generating set) {x1, …, xn} . If θ is an automorphism of Fn then {x1θ, …, xnθ} is also a basis of Fn and every basis of Fn has this form.

For any variety of groups [Vfr ], let [Vfr ](Fn) denote the verbal subgroup of Fn corresponding to [Vfr ]. (See [10] for information on varieties and related concepts.) Let Gn = Fn/[Vfr ](Fn). Then Gn is a relatively free group of rank n in the variety [Vfr ]. By a basis of Gn we mean a subset S such that every map of S into Gn extends, uniquely, to an endomorphism of Gn. Write x¯i = xi[Vfr ](Fn) for i = 1, …, n. Then {x¯1, …, x¯n} is a basis of Gn. If λ is an automorphism of Gn then {x¯1λ, …, x¯nλ} is also a basis of Gn and every basis of Gn has this form.

Any automorphism θ of Fn induces an automorphism θ of Gn in which x¯iθ = (xiθ)[Vfr ](Fn) for i = 1, …, n. Thus every basis of Fn induces a basis of Gn. The converse however is not always true; in general, there are automorphisms of Gn which are not induced by automorphisms of Fn.

If G is a finite group, X a generating set for G and R some set of relations holding among these generators (but not in general a full set of relations presenting G), we establish sufficient conditions for the group G* = 〈X/R〉 to be infinite and give a lower bound for the rank of the abelianization of the kernel of the natural map G*→G. Related results in the literature and possible directions for generalization are discussed.

Consider a closed subset of a complete Riemannian manifold, such that all geodesics with end-points in the subset are contained in the subset and the subset has boundary of codimension one. Is it the case that Riemannian barycentres of probability measures supported by the subset must also lie in the subset? It is shown that this is the case for 2-manifolds but not the case in higher dimensions: a counterexample is constructed which is a conformally-Euclidean 3-manifold, for which geodesics never self-intersect and indeed cannot turn by too much (so small geodesic balls satisfy a geodesic convexity condition), but is such that a probability measure concentrated on a single point has a barycentre at another point.

Signature invariants of odd dimensional links from irregular covers and non-abelian covers of complements are obtained by using the technique of Casson and Gordon. We show that the invariants vanish for slice links and can be considered as invariants under Fm-link concordance. We illustrate examples of links that are not slice but behave as slice links for any invariants from abelian covers.

Let f be a hyperbolic transcendental meromorphic function such that the finite singularities of f−1 are in a bounded set. We show that there exists 0<s(f)[les ]2 such that

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for each point a in the Julia set of f, where

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We then show that s(f)[les ]dimJ(f), the Hausdorff dimension of the Julia set, and give examples of such functions for which dimJ(f)>s(f). This contrasts with the situation for a hyperbolic rational function f where it is known that dimJ(f) = s(f).

In the classical Poincaré–Bendixson theory the object of study are the limit sets of a continuous flow on the 2-sphere S2 and the behaviour of the orbits near them (see [7, 9]). In [2] the second author proved that an assertion similar to the Poincaré–Bendixson theorem is true in the wider class of the 1-dimensional invariant (internally) chain recurrent continua of flows on S2. On the other hand, it is known that among the closed 2-manifolds, the 2-sphere S2, the projective plane RP2 and the Klein bottle K2 are the only ones for which the Poincaré–Bendixson theorem is true (see [1, 8, 11]).

The motivation of the present paper was to examine to what extent the main results of [2] carry over to flows on RP2 and K2. A first attempt to study chain recurrent sets of flows on closed 2-manifolds other than the 2-sphere was [3]. As one expects, the results of [2] carry over easily to RP2, since chain recurrence behaves well with respect to regular covering maps of compact manifolds, as we show in Section 3. The situation with K2 is quite different, since it is doubly covered by the 2-torus T2, where we have no Poincaré–Bendixson theorem. Actually, the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. For example, identifying suitably the boundary periodic orbits of a 2-dimensional Reeb flow on a closed annulus (see [7, chapter III, 2·6]) we get a flow on K2 with a 1-dimensional invariant chain recurrent continuum consisting of the unique periodic orbit and another orbit, which spirals against it in positive and negative time. As we prove in Theorem 4·4, this situation, or concatenations of it, is the only one where the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. Then, we are concerned with the topological structure of the 1-dimensional chain components of a flow on K2 with finitely many singularities. In Proposition 4·6 we find when such a set consists of finitely many orbits and is homeomorphic to a finite graph. An example shows that the hypothesis of Proposition 4·6 is essential. Finally, in Theorem 4·9 we give a description of the structure of the 1-dimensional chain components of a flow on K2 with finitely many singular points.