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Volume 74 Issue 03

Fri, 01 Dec 2006 00:00:00 GMT

Journal of the London Mathematical Society, Volume 74 Issue 03

Founded in 1926 and now in its Second Series, the Journal of the London Mathematical Society has a reputation for publishing some of the highest quality research on the whole spectrum of mathematics. The journal has a wide scope which ranges from number theory to functional analysis, from finite simple groups to the mathematical foundations of quantum theory and from logic and topos theory to the topology of Lie groups. In 2006, Cambridge University Press are delighted to support the London Mathematical Society in the introduction of a year's trial open access policy, in which the most recent two issues will be freely available online as part of a one year experiment. ***Latest three issues now available FREE ONLINE!***

THE ANALOGUE OF BÜCHI'S PROBLEM FOR RATIONAL FUNCTIONS

Research Articles
THANASES PHEIDAS, XAVIER VIDAUX,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 545-565

Abstract
B chi s problem asked whether there exists an integer $M$ such that the surface defined by a system of equations of the form$$x_{n}^2+x_{n-2}^2=2x_{n-1}^2+2,\quad n=2,\dotsc, M-1,$$has no integer points other than those that satisfy $\pm x_n=\pm x_0+n$ (the $\pm$ signs are independent). If answered positively, it would imply that there is no algorithm which decides, given an arbitrary system $Q=(q_1,\dotsc,q_r)$ of integral quadratic forms and an arbitrary $r$-tuple $B=(b_1,\dotsc,b_r)$ of integers, whether $Q$ represents $B$ (see T. Pheidas and X. Vidaux, Fund. Math. 185 (2005) 171 194). Thus it would imply the following strengthening of the negative answer to Hilbert s tenth problem: the positive-existential theory of the rational integers in the language of addition and a predicate for the property $x$ is a square would be undecidable. Despite some progress, including a conditional positive answer (depending on conjectures of Lang), B chi s problem remains open.In this paper we prove the following:an analogue of B chi s problem in rings of polynomials of characteristic either 0 or $p\geq17$ and for fields of rational functions of characteristic 0; andan analogue of B chi s problem in fields of rational functions of characteristic $p\geq19$, but only for sequences that satisfy a certain additional hypothesis.As a consequence we prove the following result in logic.Let $F$ be a field of characteristic either 0 or at least 17 and let $t$ be a variable. Let $L_{t}$ be the first order language which contains symbols for 0 and 1, a symbol for addition, a symbol for the property $x$ is a square and symbols for multiplication by each element of the image of $\mathbb{Z}[t]$ in $F[t]$. Let $R$ be a subring of $F(t)$, containing the natural image of $\mathbb{Z}[t]$ in $F(t)$. Assume that one of the following is true:$R\subset F[t]$;the characteristic of $F$ is either 0 or $p\geq19$.Then multiplication is positive-existentially definable over the ring $R$, in the language $L_t$. Hence the positive-existential theory of $R$ in $L_{t}$ is decidable if and only if the positive-existential ring-theory of $R$ in the language of rings, augmented by a constant-symbol for $t$, is decidable.

FINITE AXIOMATIZATION OF FINITE SOLUBLE GROUPS

Research Articles
JOHN S. WILSON,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 566-582

Abstract
It is proved that the finite soluble groups can be characterized among finite groups by a first-order sentence, namely, the sentence that asserts that no non-trivial element $g$ is a product of 56 commutators $[x,y]$ with entries $x$, $y$ conjugate to $g$.

CONSTRUCTIBLE FUNCTIONS ON ARTIN STACKS

Research Articles
DOMINIC JOYCE,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 583-606

Abstract
Let $\mathbb K$ be an algebraically closed field, let $X$ be a $\mathbb K$-variety, and let $X(\mathbb K)$ be the set of closed points in $X$. A constructible set $C$ in $X(\mathbb K)$ is a finite union of subsets $Y(\mathbb K)$ for subvarieties $Y$ in $X$. A constructible function $f:X(\mathbb K)\rightarrow\mathbb Q$ has $f(X(\mathbb K))$ finite and $f^{-1}(c)$ constructible for all $c\ne 0$. Write CF$(X)$ for the vector space of such $f$. Let $\phi:X\rightarrow Y$ and $\psi: Y\rightarrow Z$ be morphisms of ${\mathbb C}$-varieties. MacPherson defined a linear pushforward CF$(\phi):{\rm CF}(X)\rightarrow{\rm CF}(Y)$ by integration with respect to the topological Euler characteristic. It is functorial, that is, CF$(\psi\circ\phi)={\rm CF}(\psi)\circ{\rm CF}(\phi)$. This was extended to $\mathbb K$ of characteristic zero by Kennedy.This paper generalizes these results to $\mathbb K$-schemes and Artin $\mathbb K$-stacks with affine stabilizer groups. We define the notions of Euler characteristic for constructible sets in $\mathbb K$-schemes and $\mathbb K$-stacks, and pushforwards and pullbacks of constructible functions, with functorial behaviour. Pushforwards and pullbacks commute in Cartesian squares. We also define pseudomorphisms, a generalization of morphisms well suited to constructible functions problems.

DESSINS D'ENFANTS AND HYPERSURFACES WITH MANY $A_j$-SINGULARITIES

Research Articles
OLIVER LABS,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 607-622

Abstract
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.

SINGULARITIES OF ORBIT CLOSURES IN MODULE VARIETIES AND CONES OVER RATIONAL NORMAL CURVES

Research Articles
GRZEGORZ ZWARA,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 623-638

Abstract
Let $N$ be a point of an orbit closure $\overline{{\mathcal O}}_M$ in a module variety such that its orbit ${\mathcal O}_N$ has codimension 2 in $\overline{{\mathcal O}}_M$. We show that under some additional conditions the pointed variety $(\overline{{\mathcal O}}_M,N)$ is smoothly equivalent to a cone over a rational normal curve.

CO-POINT MODULES OVER KOSZUL ALGEBRAS

Research Articles
IZURU MORI,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 639-656

Abstract
Let $A$ be a graded algebra finitely generated in degree 1 over a field $k$. Point modules over $A$ introduced by Artin, Tate and Van den Bergh play an important role in studying $A$ in noncommutative algebraic geometry. In this paper, we define a dual notion of point module in terms of Koszul duality, which we call a co-point module. Using co-point modules, we will construct counter-examples to the following condition due to Auslander: for every finitely generated right module $\pi$ over a ring $R$, there is a natural number $n_M\in {\mathbb N}$ such that, for any finitely generated right module $N$ over $R$, ${\rm Ext}^i_R(M, N)=0$ for all $i\gg 0$ implies ${\rm Ext}^i_R(M, N)=0$ for all $i>n_M$.

EFFECTIVE BOUNDS ON HOLOMORPHIC MAPPINGS INTO COMPLEX HYPERBOLIC MANIFOLDS

Research Articles
JUN-MUK HWANG, WING-KEUNG TO,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 657-672

Abstract
We give an explicit upper bound on the number of non-constant holomorphic maps from a quasi-projective manifold into a complex hyperbolic manifold of finite volume. This gives an effective version of the results of Sunada and Noguchi.

POSITIVE SOLUTIONS OF NONLOCAL BOUNDARY VALUE PROBLEMS: A UNIFIED APPROACH

Research Articles
J. R. L. WEBB, GENNARO INFANTE,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 673-693

Abstract
We give a unified approach for studying the existence of multiple positive solutions of nonlinear differential equations of the form$$-u''(t)=g(t)f(t,u(t)),\quad \text{for almost every } t \in (0,1),$$where $g, f$ are non-negative functions, subject to various nonlocal boundary conditions. We study these problems via new results for a perturbed integral equation, in the space $C[0,1]$, of the form$$u(t)=\gamma(t){\alpha}[u]+\delta(t){\beta}[u]+\int_{0}^{1}k(t,s)g(s)f(s,u(s))\,ds$$where $\alpha[u]$, $\beta[u]$ are linear functionals given by Stieltjes integrals but are not assumed to be positive for all positive $u$. This means we actually cover many more differential equations than the simple equation written above. Previous results have studied positive functionals only, but even for positive functionals our methods give improvements on previous work. The well-known $m$-point boundary value problems are special cases and we obtain sharp conditions on the coefficients, which allows some of them to have opposite signs. We also use some optimal assumptions on the nonlinear term.

SEMICLASSICAL ANALYSIS FOR MAGNETIC SCATTERING BY TWO SOLENOIDAL FIELDS

Research Articles
HIROSHI T. ITO, HIDEO TAMURA,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 695-716

Abstract
That vector potentials have a direct significance to quantum particles moving in magnetic fields is known as the A B (Aharonov Bohm) effect. We study scattering by two solenoidal magnetic fields (point-like magnetic fields) in two dimensions and analyze the asymptotic behavior of the scattering amplitude in the semiclassical limit. The corresponding classical mechanical system has a trajectory oscillating between the centers of the two fields. We derive the asymptotic formula with the first three terms and make clear how such a trapping trajectory is reflected in the asymptotic formula through the A B effect. We also make a brief comment on an extension to the scattering by many solenoidal fields. The result depends on the location of the centers of the fields. In particular, the A B effect appears strongly when the centers are on an even line.

$L^p-L^q$ ESTIMATES FOR PARABOLIC SYSTEMS IN NON-DIVERGENCE FORM WITH VMO COEFFICIENTS

Research Articles
ROBERT HALLER-DINTELMANN, HORST HECK, MATTHIAS HIEBER,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 717-736

Abstract
Consider a parabolic $N\times N$-system of order $m$ on $\mathbb{R}^n$ with top-order coefficients $a_\alpha \in \mathrm{VMO} \cap L^\infty$. Let $1 and let $\omega$ be a Muckenhoupt weight. It is proved that systems of this kind possess a unique solution $u$ satisfying$$\|u'\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)} + \|\mathcal{A} u\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)} \le C \|f\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)},$$where $\mathcal{A} u = \sum_{|\alpha| \le m}a_\alpha D^\alpha u$ and $J=[0,\infty)$. In particular, choosing $\omega =1$, the realization of $\mathcal{A}$ in $L^p({\mathbb{R}}^n)^N$ has maximal $L^p-L^q$ regularity.

ABSOLUTE CONTINUITY FOR RANDOM ITERATED FUNCTION SYSTEMS WITH OVERLAPS

Research Articles
YUVAL PERES, KÁROLY SIMON, BORIS SOLOMYAK,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 739-756

Abstract
We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + \lambda_i Y x\}_{i=1}^m$, where $d_i\in {\mathbb R}$ and $\lambda_i>0$ are fixed and $Y> 0$ is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of independent and identically distributed errors $y_1,y_2,\dotsc$, distributed as $Y$, independent of everything else. Let $h$ be the entropy of the process, and let $\chi = {\mathbb E}[\log(\lambda Y)]$ be the Lyapunov exponent. Assuming that $\chi , we obtain a family of conditional measures $\nu_{\bf y}$ on the line, parametrized by ${\bf y} = (y_1,y_2,\dotsc)$, the sequence of errors. Our main result is that if $h > |\chi|$, then $\nu_{\bf y}$ is absolutely continuous with respect to the Lebesgue measure for almost every ${\bf y}$. We also prove that if $h , then the measure $\nu_{\bf y}$ is singular and has dimension $h/|\chi|$ for almost every ${\bf y}$. These results are applied to a randomly perturbed iterated function system suggested by Sinai, and to a class of random sets considered by Arratia, motivated by probabilistic number theory.

A BORG-TYPE THEOREM ASSOCIATED WITH ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE

Research Articles
FRITZ GESZTESY, MAXIM ZINCHENKO,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 757-777

Abstract
We prove a general Borg-type result for reflectionless unitary CMV operators $U$ associated with orthogonal polynomials on the unit circle. The spectrum of $U$ is assumed to be a connected arc on the unit circle. This extends a recent result of Simon in connection with a periodic CMV operator with spectrum the whole unit circle.In the course of deriving the Borg-type result we also use exponential Herglotz representations of Caratheodory functions to prove an infinite sequence of trace formulas connected with the CMV operator $U$.

REAL HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES IN COMPLEX HYPERBOLIC SPACES

Research Articles
J. BERNDT, J. C. DÍAZ-RAMOS,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 778-798

Abstract
We present the classification of all real hypersurfaces in complex hyperbolic space $\mathbb{C}H^{n}$, $n \geq 3$, with three distinct constant principal curvatures.

GEOMETRICAL SPINES OF LENS MANIFOLDS

Research Articles
S. ANISOV,
Journal of the London Mathematical Society, Volume 74 Issue 03 , pp 799-816

Abstract
We introduce the concept of geometrical spine for 3-manifolds with natural metrics, in particular, for lens manifolds. We show that any spine of $L_{p,q}$ that is close enough to its geometrical spine contains at least $E(p,q)-3$ vertices, which is exactly the conjectured value for the complexity $c(L_{p,q})$. As a byproduct, we find the minimal rotation distance (in the Sleator Tarjan Thurston sense) between a triangulation of a regular $p$-gon and its image under rotation.