good parameters for a class of node sets in quasi monte carlo integration for 2 s 12 we determine good parameters in a general construction of node sets for s dimensional quasi monte carlo integration recently introduced by the third author some of the parameters represent optimal choices in this construction and lead to improvements on node sets obtained by earlier techniques

courbes algebriques et equations multiplicatives we study the intersection of an algebraic curve c lying in a multiplicative torus over bar mathbb q with the union of all algebraic subgroups of codimension 2 finiteness of this set has already been proved by bombieri masser and zannier under the assumption that c is not contained in a translate of a proper subtorus following this result the question of the minimal hypothesis implying finiteness has been raised by these authors giving rise to the conjecture finiteness holds precisely when c is not contained in a proper subgroup we prove here this statement which is also a special case of more general conjectures stated independently by zilber and pink our proof takes its inspiration from an article by remond and viada concerning the zilber pink conjecture for curves lying in a power of an elliptic curve hence it relies on a uniform version of the vojta inequality proven via the generalized vojta inequality of remond the main task is to establish a lower bound for some intersection numbers here on a whole family of surfaces obtained by blowing up a compactification of c c

torsion points on families of squares of elliptic curves in a recent paper we proved that there are at most finitely many complex numbers 0 1 such that the points 2 sqrt 2 2 lambda and 3 sqrt 6 3 lambda are both torsion on the elliptic curve defined by y 2 x x 1 x here we give a generalization to any two points with coordinates algebraic over the field q and even over c this implies a special case of a variant of pink s conjecture for a variety inside a semiabelian scheme namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes

big line bundles over arithmetic varieties 5 equidistribution theory 26 5 1 a generic equidistribution theorem 27 5 2 equidistribution at infinite places 30 5 3 equidistribution at finite places 32 5 4 equidistribution of small subvarieties 35 5 5 equidistribution over algebraic dynamical systems 37 5 6 equidistribution over multiplicative groups 39 5 7 equidistribution theory 41

a height gap theorem for finite subsets of mathrm gl _ d overline bbb q and nonamenable subgroups we introduce a conjugation invariant normalized heightb f on nite subsets of matrices f in gld q and describe its properties in particular we prove an analogue of the lehmer problem for this height by showing that b f whenever f generates a nonvirtually solvable subgroup of gld q where d 0 is an absolute constant this can be seen as a global adelic analog of the classical margulis lemma from hyperbolic geometry as an application we prove a uniform version of the classical burnside schur theorem on torsion linear groups in a companion paper we will apply these results to prove a strong uniform version of the tits alternative

equidistribution quantitative des points de petite hauteur sur la droite projective we introduce a new class of adelic heights on the projective line we estimate their essential minimum and prove a result of equidistribution at every place for points of small height with estimates on the speed of convergence to each rational function r in one variable and defined over a number field k is associated a normalized height on the algebraic closure of k we show that these dynamically defined heights are adelic in our sense and deduce from this equidistribution results for preimages of points under r at every place of k our approach follows that of bilu and relies on potential theory in the complex plane as well as in the berkovich space associated to the projective line over open image in new window for each prime p

intersecting subvarieties of g_m n with algebraic subgroups let x subset g m n be an irreducible closed subvariety defined over bar q we bound the height of algebraic points on x that are in a certain sense close to the union of all algebraic subgroup of g m n of dimension m n dim x the bound we obtain is effective and will be expressed as a function of the height of x the degree of x and n we then apply this bound to derive certain finiteness results if m is also strictly less than n dim x

on the use of reducible polynomials as random number generators the randomness properties and the hierarchy of correlation coefficients are studied of approximate maximum length sequences for which the characteristic polynomial is a product of several primitive polynomials the randomness properties are almost the same as for maximum length sequences characterized by a primitive polynomial with many terms and of the same degree reducible characteristic polynomials have acceptable figures of merit and can be of extremely high degree since they are also easily constructed and implemented reducible polynomials are strong candidates for reliable random number generation especially at the bit rates needed in large scale monte carlo simulations

some consequences of masser s counting theorem on elliptic curves we use masser s counting theorem to prove a lower bound for the canonical height in powers of elliptic curves we also prove the galois case of the elliptic lehmer problem combining kummer theory and masser s result with bounds on the rank and torsion of some groups of rational points on an elliptic curve

calculation of fibonacci polynomials for gfsr sequences with low discrepancies fibonacci polynomials are defined in the context of the two dimensional discrepancy of tausworthe pseudorandom sequences as an analogue to fibonacci numbers which give the best figure of merit for the two dimensional discrepancy of linear congruential sequences we conduct an exhaustive search for the fibonacci polynomials of degree less than 32 whose associated tausworthe sequences can be easily implemented and very quickly generated

figures of merit for digital multistep pseudorandom numbers the statistical independence properties of s successive digital multistep pseudorandom numbers are governed by the figure of merit p s i which depends on s and the characteristic polynomial f of the recursion used in the generation procedure we extend previous work for s 2 and describe how to obtain large figures of merit for s 2 thus arriving at digital multistep pseudorandom numbers with attractive statistical independence properties tables of figures of merit for s 3 4 5 and degrees 32 are included

le probleme de lehmer en dimension superieure we study a higher dimensional lehmer problem or alternatively the lehmer problem for a power of the multiplicative group more precisely if 1 n are multiplicatively independent algebraic numbers we provide a lower bound for the product of the heights of the i s in terms of the degree d of the number field generated by the i s this enables us to study the successive minima for the height function in a given number field our bound is a generalisation of an earlier result of dobrowolski and is best possible up to a power of log d this in particular shows that the lehmer problem is true for number fields having a small galois group

intersection de sous groupes et de sous varietes i we study the intersection of a subvariety x of an abelian variety a over open image in new window with the union of all the algebraic subgroups of a of given dimension d our main result states that if we remove a suitable exceptional subset from x and if d is small enough then the intersection enjoys a northcott like property the points of bounded height on it form a finite set the condition on d involves only the dimension of x and the structure of a up to isogenies we show how it can be weakened if we assume certain conjectures in the direction of an abelian version of lehmer s problem the theorem is especially meaningful when x is a curve since it is then possible to bound the height and hence to prove finiteness of the set under consideration this generalises the result of e viada on powers of elliptic curves and is analogous to work of e bombieri d masser and u zannier on tori whose general strategy we follow

mesures et equidistribution sur les espaces de berkovich the proof by ullmo and zhang of bogomolov s conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for small points in the associated complex abelian variety we study the analogous equidistribution property at p adic places our results can be conveniently stated within the framework of the analytic spaces defined by berkovich the first one is valid in any dimension but is restricted to algebraic metrics the second one is valid for curves but allows for more general metrics in particular to the normalized heights with respect to dynamical systems

the difference between the weil height and the canonical height on elliptic curves estimates for the difference of the weil height and the canonical height of points on elliptic curves are used for many purposes both theoretical and computational in this note we give an explicit estimate for this difference in terms of the j invariant and discriminant of the elliptic curve the method of proof suggested by serge lang is to use the decomposition of the canonical height into a sum of local heights we illustrate one use for our estimate by computing generators for the mordell weil group in three examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 x ax b with a and b in the ring of integers of k the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 10 chapter viii 9 or 6 chapter vi the canonical height is determined by this property together with the fact that the difference

invariant varieties for polynomial dynamical systems we study algebraic dynamical systems and more generally s varieties f a n c a n c given by coordinatewise univariate polynomials by refining an old theorem of ritt on compositional identities amongst polynomials more precisely we find a nearly canonical way to write a polynomial as a composition of clusters from which one may easily read off possible compositional identities our main result is an explicit description of the weakly skew invariant varieties that is for a fixed field automorphism s c c those algebraic varieties x a n c for which f x x s as a special case we show that if f x c x is a polynomial of degree at least two that is not conjugate to a monomial chebyshev polynomial or a negative chebyshev polynomial and x a 2 c is an irreducible curve that is invariant under the action of x y f x f y and projects dominantly in both directions then x must be the graph of a polynomial that commutes with f under composition as consequences we deduce a variant of a conjecture of zhang on the existence of rational points with zariski dense forward orbits and a strong form of the dynamical manin mumford conjecture for liftings of the frobenius we also show that in models of acfa 0 a disintegrated set defined by s x f x for a polynomial f has morley rank one and is usually strongly minimal that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew conjugacy class of f is defined over a fixed field of a power of s and that nonorthogonality between two such sets is definable in families if the skew conjugacy class of f is defined over a fixed field of a power of s

optimal quadrature for haar wavelet spaces this article considers the error of the scrambled equidistribution quadrature rules in the worst case random case and average case settings the underlying space of integrands is a hilbert space of multidimensional haar wavelet series hwav the asymptotic orders of the errors are derived for the case of the scrambled t m s nets and t s sequences these rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands hwav

intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension in fact both authors stated stronger conjectures zilber conjectured a finiteness statement for subvarieties that are contained in an improper intersection of x with an algebraic subgroup of s pink was motivated by a more general conjecture in the context of mixed shimura varieties he also showed that conjecture 1 implies the celebrated mordell lang conjecture an important precursor was set in 1999 by bombieri masser and zannier 3 who proved the following result for an algebraic curve c embedded in the algebraic torus gm not contained in the translate of a proper algebraic subgroup and defined over q an algebraic closure of q the set of algebraic points of c which are contained in the union of all algebraic subgroups of codimension at least 1 has bounded absolute weil height they then used their result to show that there are only finitely many points on c contained in the union of all algebraic subgroups of codimension at least 2

preperiodic points and unlikely intersections in this article we combine complex analytic and arithmetic tools to study the preperiodic points of one dimensional complex dynamical systems we show that for any fixed a b c and any integer d 2 the set of c c for which both a and b are preperiodic for z c is infinite if and only if a b this provides an affirmative answer to a question of zannier which itself arose from questions of masser concerning simultaneous torsion sections on families of elliptic curves using similar techniques we prove that if rational functions c z have infinitely many preperiodic points in common then all of their preperiodic points coincide and in particular the maps must have the same julia set this generalizes a theorem of mimar who established the same result assuming that and are defined over q the main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields with non archimedean berkovich spaces playing an essential role

distribution of rational maps with a preperiodic critical point let f lambda be a family of rational maps of a fixed degree with a marked critical point c lambda under a natural assumption we first prove that the hypersurfaces of parameters for which c lambda is periodic converge as a sequence of positive closed 1 1 currents to the bifurcation current attached to c and defined by demarco we then turn our attention to the parameter space of polynomials of a fixed degree d by intersecting the d 1 currents attached to each critical point of a polynomial bassaneli and berteloot obtained a positive measure mu rm bif of finite mass which is supported on the connectedness locus they showed that its support is included in the closure of the set of parameters admitting d 1 neutral cycles we show that the support of this measure is precisely the closure of the set of strictly critically finite polynomials i e of misiurewicz points

on the distribution of k dimensional vectors for simple and combined tausworthe sequences the lattice structure of conventional linear congruential random number generators lcgs over integers is well known in this paper we study lcgs in the field of formal laurent series with coefficients in the galois field f2 the state of the generator a laurent series evolves according to a linear recursion and can be mapped to a number between 0 and 1 producing what we call a ls2 sequence in particular the sequences produced by simple or combined tausworthe generators are special cases of ls2 sequences by analyzing the lattice structure of the lcg we obtain a precise description of how all the k dimensional vectors formed by successive values in the ls2 sequence are distributed in the unit hypercube more specifically for any partition of the k dimensional hypercube into 2kl identical subcubes we can quickly compute a table giving the exact number of subcubes that contain exactly n points for each integer n we give numerical examples and discuss the practical implications of our results

representation of functions as walsh series to different bases and an application to the numerical integration of high dimensional walsh series we will prove the following theorem on walsh series and we will derive from this theorem an effective and constructive method for the numerical integration of walsh series by number theoretic methods further concrete computer calculations are given theorem for base b 2 dimension s 1 and o 1 c 0 b s n c a e r let i e c be the class of all functions f 0 l s c which are representable by absolutely convergent walsh series to base b with walsh coefficients w h hs with the following property w h hs 2 provided that a 1 s where

an enumerating function for spanning forests with color restrictions abstract consider a graph in which each edge is assigned a color a list of spanning trees with all edges of distinct colors is obtained as the formal expansion of certain mixed discriminants an enumerating function for spanning trees with prescribed number of edges of each color is derived the result generalizes the matrix tree theorem of kirchhoff in particular a list of spanning forests with k trees of a graph with n 1 vertices is obtained as the formal expansion of the sum of the principal minors of order n 1 k in the kirchhoff matrix associated to the graph

on the explicit torsion anomalous conjecture the torsion anomalous conjecture states that an irreducible variety v embedded in a semi abelian variety contains only finitely many maximal v torsion anomalous varieties in this paper we consider an irreducible variety embedded in a produc of elliptic curves our main result provides a totally explicit bound for the n eron tate height of all maximal v torsion anomalous points of relative codimension one in the non cm case and an analogous effective result in the cm case as an application we obtain the finiteness of such points in addition we deduce some new explicit results in the context of the effective mordell lang conjecture in particular we bound the n eron tate height of the rational points of an explicit family of curves of increasing genus

a multivariate fast discrete walsh transform with an application to function interpolation for high dimensional problems such as approximation and integration one cannot afford to sample on a grid because of the curse of dimensionality an attractive alternative is to sample on a low discrepancy set such as an integration lattice or a digital net this article introduces a multivariate fast discrete walsh transform for data sampled on a digital net that requires only o n log n operations where n is the number of data points this algorithm and its inverse are digital analogs of multivariate fast fourier transforms this fast discrete walsh transform and its inverse may be used to approximate the walsh coefficients of a function and then construct a spline interpolant of the function this interpolant may then be used to estimate the function s effective dimension an important concept in the theory of numerical multivariate integration numerical results for various functions are presented

forest matrices around the laplacian matrix abstract we study the matrices q k of in forests of a weighted digraph and their connections with the laplacian matrix l of the i j entry of q k is the total weight of spanning converging forests in forests with k arcs such that i belongs to a tree rooted at j the forest matrices q k can be calculated recursively and expressed by polynomials in the laplacian matrix they provide representations for the generalized inverses the powers and some eigenvectors of l the normalized in forest matrices are row stochastic the normalized matrix of maximum in forests is the eigenprojection of the laplacian matrix which provides an immediate proof of the markov chain tree theorem a source of these results is the fact that matrices q k are the matrix coefficients in the polynomial expansion of adj i l thereby they are precisely faddeev s matrices for l

torsion anomalous points and families of elliptic curves we prove that there are at most finitely many complex lambda neq 0 1 such that two points on the legendre elliptic curve y 2 x x 1 x lambda with coordinates x 2 3 both have finite order this is a very special case of some conjectures on unlikely intersections in semiabelian schemes

mixed discriminants of positive semidefinite matrices abstract if ak akij k 1 2 n are n by n matrices then their mixed discriminant d a1 an is given by d a 1 a n 1 n s n a j ij where sn is the symmetric group of degree n and where denotes determinant we give certain alternative ways of defining the mixed discriminant and state some basic properties it is pointed out that a ryser type formula for the mixed discriminant exists in the literature and a simpler proof is given for it it is shown that the mixed discriminant can be expressed as an inner product a generalization of konig s theorem on 0 1 matrices is proved the following set d n which includes the set of n by n doubly stochastic matrices is defined and studied

bifurcation measures and quadratic rational maps we study critical orbits and bifurcations in the curves per1 within the moduli space m2 of quadratic rational maps f p 1 p 1 for each 2 c the curve per1 is dened by the condition that the map has a xed point of multiplier we prove that the curve per1 contains innitely many postcritically n ite maps if and only if 0 addressing a special case of bd2 conjecture 1 4 we also show that the two critical points of f dene distinct bifurcation measures along per 1

the arithmetic hodge index theorem for adelic line bundles in this paper we prove index theorems for integrable metrized line bundles on projective varieties over complete fields and number fields respectively as applications we prove a non archimedean analogue of the calabi theorem and a rigidity theorem about the preperiodic points of algebraic dynamical systems

optimal polynomials for t m s nets and numerical integration of multivariate walsh series existence theorems for good parameters in a construction of low discrepancy point sets based on rational functions over finite fields are proved a special class of these point sets is introduced and good parameters in the construction of these special point sets are tabulated in this way point sets in the s dimensional unit cube for 3 leq s leq 15 are obtained with very small discrepancy these special point sets yield excellent numerical results when applied to the quasi monte carlo integration of multivariate walsh series

intersecting curves and algebraic subgroups conjectures and more results this paper solves in the affirmative up to dimension n 5 a question raised in an earlier paper by the authors the equivalence of the problem with a conjecture of shou wu zhang is proved in the appendix

torsion points and the lattes family we give a dynamical proof of a result of masser and zannier for any a neq b in overline bbb q setminus 0 1 there are only finitely many parameters t in bbb c for which points p a a sqrt a a 1 a t and p b b sqrt b b 1 b t are both torsion on the legendre elliptic curve e t y 2 x x 1 x t our method also gives the finiteness of parameters t where both p a and p b have small n eron tate height a key ingredient in the proof is an arithmetic equidistribution theorem on bbb p 1 for this we prove two statements about the degree 4 latt es family f t on bbb p 1 1 for each c in bbb c t the bifurcation measure mu c for the pair f t c has continuous potential across the singular parameters t 0 1 infty and 2 for distinct points a b in bbb c setminus 0 1 the bifurcation measures mu a and mu b cannot coincide combining our methods with the result of masser and zannier we extend their conclusion to points t of small height also for a b in bbb c t

on the numerical integration of walsh series by number theoretic methods in analogy to the theory of good lattice points for the numerical integration of rapidly converging fourier series a theory for the fast numerical integration of walsh series is developed the basis for this theory is provided by a class of very well distributed point sets in the s dimensional unit cube the so called t m s nets

finiteness of hilbert functions and bounds for castelnuovo mumford regularity of initial ideals bounds for the castelnuovo mumford regularity and hilbert coefficients are given in terms of the arithmetic degree if the ring is reduced or in terms of the defining degrees from this it follows that there exists only a finite number of hilbert functions associated with reduced algebras over an algebraically closed field with a given arithmetic degree and dimension a good bound is also given for the castelnuovo mumford regularity of initial ideals which depends neither on term orders nor on the coordinates and holds for any field

dynamical anomalous subvarieties structure and bounded height theorems abstract according to medvedev and scanlon 14 a polynomial f x q x of degree d 2 is called disintegrated if it is not conjugate to x d or to c d x where c d is the chebyshev polynomial of degree d let n n let f 1 f n q x be disintegrated polynomials of degrees at least 2 and let f 1 f n be the corresponding coordinate wise self map of p 1 n let x be an irreducible subvariety of p 1 n of dimension r defined over q we define the anomalous locus of x which is related to the periodic subvarieties of p 1 n we prove that the anomalous locus of x is zariski closed this is a dynamical analogue of a theorem of bombieri masser and zannier 4 we also prove that the points in the intersection of x with the union of all irreducible periodic subvarieties of p 1 n of codimension r have bounded height outside the anomalous locus of x this is a dynamical analogue of habegger s theorem 8 which was previously conjectured in 4 the slightly more general self maps f 1 f n where each f i q x is a disintegrated rational function are also treated at the end of the paper

critical orbits of polynomials with a periodic point of specified multiplier answering a question posed by adam epstein we show that the collection of conjugacy classes of polynomials admitting a parabolic fixed point and at most one infinite critical orbit is a set of bounded height in the relevant moduli space we also apply the methods over function fields to draw conclusions about algebraically parametrized families and prove an analogous result for quadratic rational maps

randomized polynomial lattice rules for multivariate integration and simulation lattice rules are among the best methods to estimate integrals in a large number of dimensions they are part of the quasi monte carlo set of tools a theoretical framework for a class of lattice rules defined in a space of polynomials with coefficients in a finite field is developed in this paper a randomized version is studied implementations and criteria for selecting the parameters are discussed and examples of its use as a variance reduction tool in stochastic simulation are provided certain types of digital net constructions as well as point sets constructed by taking all vectors of successive output values produced by a tausworthe random number generator are special cases of this method