a parametric family of quintic thue equations for an integral parameter t z we investigate the family of thue equations f x y x 5 t 1 2 x 4 y 2t 3 4t 4 x 3 y 2 t 4 t 3 2t 2 4t 3 x 2 y 3 t 3 t 2 5t 3 xy 4 y 5 1 originating from emma lehmer s family of quintic fields and show that for t 3 28 10 15 the only solutions are the trivial ones with x 0 or y 0 our arguments contain some new ideas in comparison with the standard methods for thue families which gives this family a special interest

on the representation of unity by binary cubic forms if f x y is a binary cubic form with integer coefficients such that f x 1 has at least two distinct complex roots then the equation f x y 1 possesses at most ten solutions in integers x and y nine if f has a nontrivial automorphism group if further f x y is reducible over z x y then this equation has at most 2 solutions unless f x y is equivalent under gl2 z action to either x x2 xy y2 or x x2 2y2 the proofs of these results rely upon the method of thue siegel as refined by evertse together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational diophantine approximation along the way we completely solve all thue equations f x y 1 for f cubic and irreducible of positive discriminant df 106 as corollaries we obtain bounds for the number of solutions to more general cubic thue equations of the form f x y m and to mordell s equation y2 x3 k where m and k are nonzero integers

kahler einstein submanifolds of the infinite dimensional projective space this paper consists of two main results in the first one we describe all kahler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the wallach set of the domain in the second one we exhibit an example of complete and non homogeneous kahler einstein metric with negative scalar curvature which admits a kahler immersion into the infinite dimensional complex projective space

non algebraic integrability of the chew low reversible dynamical system of the cremona type and the relation with the 7th hilbert problem non resonant case abstract a smooth reversible dynamical system srds and a system of nonlinear functional equations defined by a certain rational quadratic cremona mapping and arising from the static model of the dispersion approach in the theory of strong interactions the chew low equations for p wave n scattering are considered this srds is splitted into 1 and 2 dimensional ones an explicit cremona transformation that completely determines the exact solution of the two dimensional system is found this solution depends on an odd function satisfying a nonlinear autonomous 3 point functional equation non algebraic integrability of srds under consideration is proved using the method of poincare normal forms and the siegel theorem on biholomorphic linearization of a mapping at a non resonant fixed point the proof is based on the classical feldman baker theorem on linear forms of logarithms of algebraic numbers which in turn relies upon solving the 7th hilbert problem by a i gel fond and t schneider and new powerful methods of a baker in the theory of transcendental numbers the general theorem following from the feldman baker theorem on applicability of the siegel theorem to the set of the eigenvalues c n of a mapping at a non resonant fixed point which belong to the algebraic number field a is formulated and proved the main results are presented in theorems 1 3 5 7 8 and remarks 3 7

on the linear independence measure of logarithms of rational numbers in this paper we give a general theorem on the linear independence measure of logarithms of rational numbers and in particular the linear independence measure of 1 log 2 log 3 log 5 and of 1 log 2 log 3 log 5 log 7 we also give a method to search for polynomials of smallest norm on a real interval a b which may be suitable for computing or improving the linear independence measure of logarithms of rational numbers

on the abc conjecture ii let x y and z be coprime positive integers with x y z in this paper we give upper bounds for z in terms of the greatest square free factor of xyz

solving resultant form equations over number fields we give an efficient algorithm for solving resultant form equations over number fields this is the first time that such equations are completely solved by reducing them to unit equations in two variables

on the solutions of a family of quartic thue equations in this paper we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields in fact we prove that for n 5 10 6 and n n 1 191 x 10 19 with n n 2 n 2 4 square free the thue equation n x y x 4 n 2 x 3 y n 3 2n 2 4n 2 x 2 y 2 n 2 xy 3 y 4 1 has no integral solution except the trivial ones 1 0 1 0 0 1 0 1

there are only finitely many diophantine quintuples a set of m positive integers is called a diophantine m tuple if the product of its any two distinct elements increased by 1 is a perfect square diophantus found a set of four positive rationals with the above property the first diophantine quadruple was found by fermat the set 1 3 8 120 baker and davenport proved that this particular quadruple cannot be extended to a diophantine quintuple in this paper we prove that there does not exist a diophantine sextuple and that there are only finitely many diophantine quintuples

graph matching with a dual step em algorithm this paper describes a new approach to matching geometric structure in 2d point sets the novel feature is to unify the tasks of estimating transformation geometry and identifying point correspondence matches unification is realized by constructing a mixture model over the bipartite graph representing the correspondence match and by affecting optimization using the em algorithm according to our em framework the probabilities of structural correspondence gate contributions to the expected likelihood function used to estimate maximum likelihood transformation parameters these gating probabilities measure the consistency of the matched neighborhoods in the graphs the recovery of transformational geometry and hard correspondence matches are interleaved and are realized by applying coupled update operations to the expected log likelihood function we evaluate the technique on two real world problems

balanced metrics on cartan and cartan hartogs domains this paper consists of two results dealing with balanced metrics in donaldson terminology on noncompact complex manifolds in the first one we describe all balanced metrics on cartan domains in the second one we show that the only cartan hartogs domain which admits a balanced metric is the complex hyperbolic space by combining these results with those obtained in loi and zedda mathematische annalen 2011 to appear we also provide the first example of complete kahler einstein and projectively induced metric g such that g is not balanced for all 0

on the diophantine equation ax n by n if a b and n are positive integers with b a and n 3 then the equation of the title possesses at most one solution in positive integers x and y with the possible exceptions of a b n satisfying b a 1 2 a min 0 3n 83 and 17 n 347 the proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometric functions the theory of linear forms in logarithms and recent computational methods related to lattice basis reduction additionally we compare and contrast a number of these last mentioned techniques

the mixed littlewood conjecture for pseudo absolute values in this paper we study the mixed littlewood conjecture with pseudo absolute values we show that if p is a prime and mathcal d is a pseudo absolute value sequence satisfying mild conditions then begin aligned inf n in mathbb n n n p n mathcal d vert n alpha vert 0 quad text for text all alpha in mathbb r end aligned our proof relies on a measure rigidity theorem due to lindenstrauss and lower bounds for linear forms in logarithms due to baker and wustholz we also deduce the answer to the related metric question of how fast the infimum above tends to zero for almost every alpha

primitive divisors of lucas and lehmer sequences stewart reduced the problem of determining all lucas and lehmer sequences whose nth element does not have a primitive divisor to solving certain thue equations using the method of tzanakis and de weger for solving thue equations we determine such sequences for n 30 further computations lead us to conjecture that for n 30 the nth element of such sequences always has a primitive divisor

classical and modular approaches to exponential diophantine equations i fibonacci and lucas perfect powers this is the first in a series of papers whereby we combine the classical approach to exponential diophantine equations linear forms in logarithms thue equations etc with a modular approach based on some of the ideas of the proof of fermat s last theorem in this paper we give new improved bounds for linear forms in three logarithms we also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the fibonacci sequence are 0 1 8 and 144 and the only perfect powers in the lucas sequence are 1 and 4

two conjectures on ricci flat kahler metrics we propose two conjectures about ricci flat kahler metrics conjecture 1 a ricci flat projectively induced metric is flat conjecture 2 a ricci flat metric on an n dimensional complex manifold such that the a n 1 coefficient of the tyz expansion vanishes is flat we verify conjecture 1 see theorem 1 1 under the assumptions that the metric is radial and stable projectively induced and conjecture 2 see theorem 1 2 for complex surfaces whose metric is either radial or complete and ale we end the paper by showing by means of the simanca metric that the assumption of ricci flatness in conjecture 1 and in theorem 1 2 cannot be weakened to scalar flatness see theorem 1 3

on the resolution of relative thue equations an efficient algorithm is given for the resolution of relative thue equations the essential improvement is the application of an appropriate version of wildanger s enumeration procedure based on the ellipsoid method of fincke and pohst recently relative thue equations have gained an important application e g in computing power integral bases in algebraic number fields the presented methods can surely be used to speed up those algorithms the method is illustrated by numerical examples

on holomorphic isometric immersions of nonhomogeneous kahler einstein manifolds into the infinite dimensional complex projective space abstract in the paper we study the existence of holomorphic isometric immersions from nonhomogeneous kahler einstein manifolds into infinite dimensional complex projective space it can also be regarded as an application of explicit solutions of complex monge ampere equations on some pseudoconvex domains

the normal holonomy group of kahler submanifolds we study the restricted holonomy group rm hol nabla perp of the normal connection nabla perp shortened to normal holonomy group of a kahler submanifold of a complex space form we prove that if the normal holonomy group acts irreducibly on the normal space then it is linear isomorphic to the holonomy group of an irreducible hermitian symmetric space in particular it is a compact group and the complex structure j belongs to its lie algebra we prove that the normal holonomy group acts irreducibly if the submanifold is full that is it is not contained in a totally geodesic proper kahler submanifold and the second fundamental form at some point has no kernel for example a kahler einstein submanifold of mathbb c p n has this property we define a new invariant mu of a kahler submanifold of a complex space form for non full submanifolds the invariant mu measures the deviation of j from belonging to the normal holonomy algebra for a kahler einstein submanifold the invariant mu is a rational function of the einstein constant by using the invariant mu we prove that the normal holonomy group of a not necessarily full kahler einstein submanifold of mathbb c p n is compact and we give a list of possible holonomy groups the approach is based on a definition of the holonomy algebra rm hol p of an arbitrary curvature tensor field p on a vector bundle with a connection and on a de rham type decomposition theorem for rm hol p

p adic logarithmic forms and a problem of erdős for any natural number m 1 let p m denote the greatest prime divisor of m by the problem of erd s in the title of the present paper we mean the general version of his problem that is his conjecture from 1965 that frac p 2 n 1 n to infty quad rm as n to infty see erd s 10 and its far reaching generalization to lucas and lehmer numbers in the present paper the author delivers three refinements upon yu 40 required by c l stewart for solving completely the problem of erd s see stewart 25 the author gives also some remarks on the solution of this problem aiming to be more streamlined with respect to the p adic theory of logarithmic forms

the explicit solutions for a class of complex monge ampere equations abstract one of the purposes of the present paper is to point out the efficiency of an approach to a class of complex monge ampere equations which are related to kahler einstein metrics by this method we obtain all the strictly plurisubharmonic solutions of the monge ampere equations on a class of domains

labeled point pattern matching by delaunay triangulation and maximal cliques abstract in many problems of pattern recognition objects in an image may be efficiently represented by a set of labeled points this paper describes a method for matching labeled point patterns with one of the prototypes the delaunay triangulation is used to partition a point pattern into a set of triangles and the largest maximal clique of the consistency graph for each triangle is used to obtain the largest set of mutually consistent point pairs the proposed method is invariant under affine transformation of point patterns and allows for additions and deletions of points and some random perturbations in their relative locations the method is applied to the matching of constellations

a toric extension of faltings diophantine approximation on abelian varieties for divisors on abelian varieties faltings established an optimal bound on the proximity of rational points to the same we extend this both to the quasiprojective category where the role of abelian varieties is played by their toroidal extensions and to holomorphic maps from the line proving along the way some wholly general dynamic intersection estimates in value distribution theory of independent interest

a thue equation with quadratic integers as variables in an earlier paper we determined all the solutions in z of a cubic thue equation with coefficients in a quadratic number field it is now shown that the method used there can be used to solve the more general problem of determining all the solutions of the thue equation in a ring of quadratic integers

properties of squeezing functions and global transformations of bounded domains the central purpose of the present paper is to study boundary behavior of squeezing functions on bounded domains we prove that the squeezing function of a strongly pseudoconvex domain tends to 1 near the boundary in fact such an estimate is proved for the squeezing function on any domain near its globally strongly convex boundary points we also study the stability of squeezing functions on a sequence of bounded domains and give comparisons of intrinsic measures and metrics on bounded domains in terms of squeezing functions as applications we give new and simple proofs of several well known results about geometry of strongly pseudoconvex do mains and prove that all cartan hartogs domains are homogenous regular finally some related problems that ask for further study are proposed

the development and comparison of robust methodsfor estimating the fundamental matrix this paper has two goals the first is to develop a variety of robust methods for the computation of the fundamental matrix the calibration free representation of camera motion the methods are drawn from the principal categories of robust estimators viz case deletion diagnostics m estimators and random sampling and the paper develops the theory required to apply them to non linear orthogonal regression problems although a considerable amount of interest has focussed on the application of robust estimation in computer vision the relative merits of the many individual methods are unknown leaving the potential practitioner to guess at their value the second goal is therefore to compare and judge the methods comparative tests are carried out using correspondences generated both synthetically in a statistically controlled fashion and from feature matching in real imagery in contrast with previously reported methods the goodness of fit to the synthetic observations is judged not in terms of the fit to the observations per se but in terms of fit to the ground truth a variety of error measures are examined the experiments allow a statistically satisfying and quasi optimal method to be synthesized which is shown to be stable with up to 50 percent outlier contamination and may still be used if there are more than 50 percent outliers performance bounds are established for the method and a variety of robust methods to estimate the standard deviation of the error and covariance matrix of the parameters are examined the results of the comparison have broad applicability to vision algorithms where the input data are corrupted not only by noise but also by gross outliers

on the complexity of the orbit problem we consider higher dimensional versions of kannan and lipton s orbit problem determining whether a target vector space may be reached from a starting point x under repeated applications of a linear transformation a answering two questions posed by kannan and lipton in the 1980s we show that when has dimension one this problem is solvable in polynomial time and when has dimension two or three the problem is in nprp

object pose the link between weak perspective paraperspective and full perspective recently dementhon and davis 1992 1995 proposed a method for determining the pose of a 3 d object with respect to a camera from 3 d to 2 d point correspondences the method consists of iteratively improving the pose computed with a weak perspective camera model to converge at the limit to a pose estimation computed with a perspective camera model in this paper we give an algebraic derivation of dementhon and davis method and we show that it belongs to a larger class of methods where the perspective camera model is approximated either at zero order weak perspective or first order paraperspective we describe in detail an iterative paraperspective pose computation method for both non coplanar and coplanar object points we analyse the convergence of these methods and we conclude that the iterative paraperspective method proposed in this paper has better convergence properties than the iterative weak perspective method we introduce a simple way of taking into account the orthogonality constraint associated with the rotation matrix we analyse the sensitivity to camera calibration errors and we define the optimal experimental setup with respect to imprecise camera calibration we compare the results obtained with this method and with a non linear optimization method

a note on divisible points of curves let c be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3 we partially answer a question posed by levin on points on c for which a non trivial power lies again on c our results have connections to zilber s conjecture on intersections with tori and yield to methods arising in transcendence theory and the theory of o minimal structures

3 d pose from 3 points using weak perspective this correspondence discusses computing the pose of a model from three matching point pairs under weak perspective projection a new approach to the problem that is motivated geometrically is described like previous methods the method here involves solving a biquadratic equation but here the biquadratic s solutions comprised of an actual and a false solution are interpreted graphically the final equations take a new form which leads to a simple expression for the image position of any unmatched model point

solving thue equations without the full unit group the main problem when solving a thue equation is the computation of the unit group of a certain number field it is shown that the knowledge of a subgroup of finite index is in fact sufficient two examples linked with the primitive divisor problem for lucas and lehmer sequences are given la principale difficulte lors de la resolution d une equation de thue reside dans le calcul du groupe des unites du corps de nombres associe on montre qu il suffit en fait de connaitre un sous groupe d indice fini de ce groupe on donne deux exemples li

index theory for toeplitz operators on bounded symmetric domains in this note we give an index theory for toeplitz operators on the hardy space of the shilov boundary of an arbitrary bounded symmetric domain our results generalize earlier work of gohberg krein and venugopalkrishna 12 for domains of rank 1 and of berger coburn koranyi 1 for domains of rank 2 bounded symmetric domains cartan domains classical or exceptional are the natural higher dimensional analogues of the open unit disk each such domain is homogeneous under a semisimple lie group of biholomorphic transformations and has an essentially unique realization as a convex circular domain which is conveniently described in jordan algebraic terms the underlying vector space z c n carries a jordan triple product denoted by uyv w uv w and the bounded symmetric domain d is the open unit ball of z with respect to the spectral norm 7 the basic example is the space z c of rectangular matrices with jordan triple product uv w uv w wv u 2 giving rise to the hyperbolic matrix ball d z g z spectrum z z 1 the symmetric domains of rank 1 are the smooth hilbert balls in c n for domains d of higher rank r the boundary dd forms a nonsmooth stratified space in jordan algebraic terms the structure of dd can be described as follows an element e z satisfying ee e e is called a tripotent for matrices the tripotents are just the partial isometries every tripotent e induces a splitting of z into the peirce spaces z e z g z ee z xz for a 0 1 the set

function spaces and reproducing kernels on bounded symmetric domains there are several natural hilbert spaces of holomorphic functions on a bounded symmetric domain such are the bergman type spaces on which the holomorphic discrete series operates and the hardy type spaces which are related to the analytic continuation of the holomorphic discrete series also closely related is the bergmann space of entire functions on the ambient which arises as the closure of the polynomials with respect to a natural inner product the space of holomorphic polynomials decomposes into irreducible subspaces under the action of the isotropy group k of the domain the main facts about this decomposition were proved by schmid 21 for another proof see 22 each irreducible subspace contains a unique normalized l invariant spherical polynomial where l is the isotropy group of the shilov boundary in k our first main result is the explicit computation of the norms of the spherical polynomials with respect to each of the hilbert spaces considered for the domains of classical type they were considered by hua 9 and for some of the hilbert spaces this was done before by upmeier 24 using different methods for certain others there are partial results in 22 we are able to do this in a fairly simple unified way by making strong use of gindikin s generalized gamma function s next we obtain a description of the reproducing kernels of the k irreducible subspaces in each of our hilbert spaces and an expansion in terms of these for

the solution of triangularly connected decomposable form equations an algorithm is given to solve the equations of the title it generalizes an earlier algorithm to solve discriminant form equations an application is given to finding curves of genus 2 with good reduction outside a given finite set of primes and weierstrass points in given number fields

distribution of integral division points on the algebraic torus let k be a number field with algebraic closure overline k and let s be a finite set of places of k containing all the infinite ones let it gamma 0 be a finitely generated subgroup of mathbb g textup m overline k and let it gamma subset mathbb g textup m overline k be the division group attached to it gamma 0 here is an illustration of what we will prove in this article fix a proper closed subinterval i of 0 infty and a nonzero effective divisor d on mathbb g textup m which is not the translate of any torsion divisor on the algebraic torus mathbb g textup m by any point of it gamma with height belonging to i then we prove a statement which easily implies that the set of integral division points on mathbb g textup m with height near i i e the set of points of it gamma with standard absolute logarithmic weil height in j which are s integral on mathbb g textup m relative to d is finite for some fixed subinterval j of 0 infty properly containing i we propose a conjecture on the nondensity of integral division points on semi abelian varieties with prescribed height values which generalizes some previously known conjectures as well as this finiteness result for mathbb g textup m finally we also propose an analogous version for a dynamical system on mathbb p 1