Let $f\in \mathbb{Z}[ x,y]$ be a reducible homogeneous polynomial of degree 3. We show that $f(x,y)$ has an even number of prime factors as often as an odd number of prime factors.

It is proved that an excellent ring which is the homomorphic image of a regular ring of finite Krull dimension and having a resolution of singularities satisfies the uniform Artin–Rees property. Extensions of some results of Lipman, Sathaye and Teissier are also obtained.

A few steps are made towards a representation theory of embeddability among uncountable graphs. A class of graphs is defined by forbidding some countable configurations which are related to the graph's end-structure. Using uncountable combinatorics, a representation theorem for embeddability in this class is proved, which asserts the existence of a surjective homomorphism from the relation of embeddability over isomorphism types of regular cardinality λ>[Nfr ]1 onto set inclusion over all subsets of reals or cardinality λ or less. As corollaries we obtain: (1) the complexity of the class in every regular uncountable λ>[Nfr ]1 is at least

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(2) a characterisation of graphs in the class for which some invariant of the graph has to be inherited by one of fewer than λ subgraphs whose union covers G. Some continuity properties of the homomorphism in the representation theorem are explored and are used to extend the first corollary to all singular cardinals below the first fixed point of second order. The first corollary shows that, contrary to what Shelah has shown for the class of all graphs, the relations of embeddability in the class under discussion is not independent of negations of the generalised continuum hypothesis.

Several differential geometric aspects of the space of representations of amenable Banach algebras are studied: differential structure, natural connection, geodesics, reductive structure, and so on. As an application we get a characterisation of amenable groups in terms of the existence of a reductive structure in that space.

We show the existence of surfaces of degree $d$ in ${\mathbb P}^3({\mathbb C})$ with approximately $(3j+2)/(6j(j+1))\,d^3$ singularities of type $A_j, 2\le j\le d-1$. The result is based on Chmutov's construction of nodal surfaces. For the proof we use plane trees related to the theory of Dessins d'Enfants.

Our examples improve the previously known lower bounds for the maximum number $\mu_{A_j}(d)$ of $A_j$-singularities on a surface of degree $d$ in most cases. We also give a generalization to higher dimensions which leads to new lower bounds even in the case of nodal hypersurfaces in ${\mathbb P}^n, n\geq5$.

To conclude, we work out in detail a classical idea of Segre which leads to some interesting examples, for example, to a sextic with 36 cusps.

Let [gfr ] be a semisimple Lie algebra over an algebraically closed field [ ] of characteristic zero and G be its adjoint group. The notion of a principal nilpotent pair is a double counterpart of the notion of a regular (= principal) nilpotent element in [gfr ]. Roughly speaking, a principal nilpotent pair e = (e1, e2) consists of two commuting elements in [gfr ] that can independently be contracted to the origin and such that their simultaneous centralizer has the minimal possible dimension, that is, rk[gfr ]. The definition and the basic results are due to V. Ginzburg [3]. He showed that the theory of principal nilpotent pairs yields a refinement of well-known results by B. Kostant on regular nilpotent elements in [gfr ] and has interesting applications to representation theory. In particular, he proved that the number of G-orbits of principal nilpotent pairs is finite and gave a classification for [gfr ] = [sfr ][lfr ]([ ]). Trying to achieve a greater generality, Ginzburg also introduced a wider class of distinguished nilpotent pairs and, again, classified them for [sfr ][lfr ]([ ]). (The precise definitions for all notions related to nilpotent pairs are found in §1.)

It is proved that the associated variety of a Poisson prime ideal of the centre of a symplectic reflection algebra at parameter $t \,{=}\, 0$ is irreducible.

With probabilistic techniques, it is proved that an inner function in the little Bloch space can have every asymptotic radial behaviour on sets of Hausdorff dimension 1. This is applied to deduce some differentiability properties of positive singular smooth measures and a result on radial cluster set interpolation for inner functions.

Let X be a locally compact metric absolute neighbourhood retract for metric spaces, U ⊂ X be an open subset and f: U → X be a continuous map. The aim of the paper is to study the fixed point index of the map that f induces in the hyperspace of X. For any compact isolated invariant set, K ⊂ U, this fixed point index produces, in a very natural way, a Conley-type (integer valued) index for K. This index is computed and it is shown that it only depends on what is called the attracting part of K. The index is used to obtain a characterization of isolating neighbourhoods of compact invariant sets with non-empty attracting part. This index also provides a characterization of compact isolated minimal sets that are attractors.

The paper shows the existence of several new families of noncompact constant mean curvature surfaces: (i) singly punctured surfaces of arbitrary genus $g\,{\geq}\,1$, (ii) doubly punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.

Let A be a Banach algebra, and let E be a Banach A-bimodule. A linear map S:A→E is intertwining if the bilinear map

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is continuous, and a linear map D[ratio ]A→E is a derivation if δ1D=0, so that a derivation is an intertwining map. Derivations from A to E are not necessarily continuous.

The purpose of the present paper is to prove that the continuity of all intertwining maps from a Banach algebra A into each Banach A-bimodule follows from the fact that all derivations from A into each such bimodule are continuous; this resolves a question left open in [1, p. 36]. Indeed, we prove a somewhat stronger result involving left- (or right-) intertwining maps.

In Merel's recent proof [7] of the uniform boundedness conjecture for the torsion of elliptic curves over number fields, a key step is to show that for sufficiently large primes N, the Hecke operators T1, T2, …, TD are linearly independent in their actions on the cycle e from 0 to i∞ in H1(X0(N) (C), Q). In particular, he shows independence when max(D8, 400D4) < N/(log N)4. In this paper we use analytic techniques to show that one can choose D considerably larger than this, provided that N is large.

three families of pairs of curves are presented; each pair consists of geometrically non-isomorphic curves whose jacobians are isomorphic to one another as unpolarized abelian varieties. each family is parametrized by an open subset of ${\mathbb p}^1$. the first family consists of pairs of genus-2 curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible jacobians. the second family also consists of pairs of genus-2 curves, but generically the curves in this family have absolutely simple jacobians. the third family consists of pairs of genus-3 curves, one member of each pair being a hyperelliptic curve and the other a plane quartic. examples from these families show that in general it is impossible to tell from the jacobian of a genus-2 curve over $\mathbb q$ whether or not the curve has rational points – or indeed whether or not it has real points. the families are constructed using methods that depend on earlier joint work with franck leprévost and bjorn poonen, and on peter bending's explicit description of the curves of genus 2 whose jacobians have real multiplication by $\mathbb z[\sqrt{2}]$.

If σ is an automorphism and δ is a σ-derivation of a ring R, then the subring of invariants is the set R(δ)={r∈R[mid ]δ(r)=0}. The main result of this paper is ‘let R be a semiprime ring with an algebraic σ-derivation δ such that R(δ) is central; then R is commutative’. This theorem generalizes results on the invariants of automorphisms and derivations and is proved by reducing down to the special cases of automorphisms and derivations.

The classical Ramsey theorem states that every graph contains either a large clique or a large independent set. Here similar dichotomic phenomena are investigated in the context of finite metric spaces. Namely, statements are provided of the form ‘every finite metric space contains a large subspace that is nearly equilateral or far from being equilateral’. Two distinct interpretations are considered for being ‘far from equilateral’. Proximity among metric spaces is quantified through the metric distortion $\alpha$. Tight asymptotic answers are provided for these problems. In particular, it is shown that a phase transition occurs at $\alpha=2$.

The defining ideal $I_{\bb X}$ of a set of points $\bb{X}$ in $\bb{P}^n_1 x \ldots x \bb{P}n_k$ is investigated with a special emphasis on the case when $\bb{X}$ is in generic position, that is, $\bb{X}$ has the maximal Hilbert function. When $\bb{X}$ is in generic position, the degrees of the generators of the associated ideal $I_{\bb X}$ are determined. $\nu(I_{\bb X})$ denotes the minimal number of generators of $I_{\bb X}$, and this description of the degrees is used to construct a function $v(s;n_1,\ldots,n_k)$ with the property that $\nu(I_{\bb X})\,{\geq}\, v(s;n_1,\ldots,n_k)$ always holds for $s$ points in generic position in $\bb{P}n_1 x \ldots x \bb{P}^n_k$. When $k\,{=}\,1$, $v(s;n_1)$ equals the expected value for $\nu(I_{\bb X})$ as predicted by the ideal generation conjecture. If $k\,{\geq}\, 2$, it is shown that there are cases with $\nu(I_{\bb X}) > v(s;n_1,\ldots,n_k$). However, computational evidence suggests that in many cases $\nu(I_{\bb X})\,{=}\, v(s;n_1,\ldots,n_k$).

The aim of the paper is to establish a complete scale of isomorphisms for elliptic pseudodifferential boundary value problems generated by the operators of the Boutet de Monvel algebra. The result is an extension of the corresponding known results on elliptic differential boundary value problems. Because for any elliptic pseudodifferential boundary value problem there exists a parametrix belonging to the Boutet de Monvel algebra, the proof presented is much shorter than the known proof for differential problems. Systems that are elliptic in the sense of Douglis and Nirenberg are also considered.

The cohomology of [Mscr ](n, d), the moduli space of stable holomorphic bundles of coprime rank n and degree d and fixed determinant, over a Riemann surface Σ of genus g [ges ] 2, has been widely studied from a wide range of approaches. Narasimhan and Seshadri [17] originally showed that the topology of [Mscr ](n, d) depends only on the genus g rather than the complex structure of Σ. An inductive method to determine the Betti numbers of [Mscr ](n, d) was first given by Harder and Narasimhan [7] and subsequently by Atiyah and Bott [1]. The integral cohomology of [Mscr ](n, d) is known to have no 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 relations that hold amongst these generators, particularly in the rank two, odd degree case which is now largely understood. A set of relations due to Mumford in the rational cohomology ring of [Mscr ](2, 1) is now known to be complete [14]; recently several authors have found a minimal complete set of relations for the ‘invariant’ subring of the rational cohomology of [Mscr ](2, 1) [2, 13, 20, 25].

Unless otherwise stated all cohomology in this paper will have rational coefficients.

Suppose that f(z) is a meromorphic function of bounded characteristic in the unit disk Δ[ratio ][mid ]z[mid ]<1. Then we shall say that f(z)∈N. It follows (for example from [3, Lemma 6.7, p. 174 and the following]) that

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where h1(z), h2(z) are holomorphic in Δ and have positive real part there, while Π1(z), Π2(z) are Blaschke products, that is,

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where p is a positive integer or zero, 0<[mid ]aj[mid ]<1, c is a constant and

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We note in particular that, if c≠0, so that f(z)[nequiv ]0,

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so that f(z)=0 only at the points aj. Suppose now that zj is a sequence of distinct points in Δ such that

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If f(zj)=0 for each j and f∈N, then f(z)≡0.

N. Danikas [1] has shown that the same conclusion obtains if f(zj)→0 sufficiently rapidly as j→∞. Let εj, λj be sequences of positive numbers such that

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Danikas then defines

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and proves Theorem A.

A geometric property is introduced, and it is proved that Banach spaces with this property must carry weakly amenable algebras of approximable operators. These results are also applied to particular examples of Banach spaces, including the James spaces and the Tsirelson space.