local spectral gap in simple lie groups and applications we introduce a novel notion of local spectral gap for general possibly infinite measure preserving actions we establish local spectral gap for the left translation action gamma curvearrowright g whenever gamma is a dense subgroup generated by algebraic elements of an arbitrary connected simple lie group g this extends to the non compact setting works of bourgain and gamburd invent math 171 83 121 2008 j eur math soc jems 14 1455 1511 2012 and benoist and de saxce invent math 205 337 361 2016 we present several applications to the banach ruziewicz problem orbit equivalence rigidity continuous and monotone expanders and bounded random walks on g in particular we prove that up to a multiplicative constant the haar measure is the unique gamma invariant finitely additive measure defined on all bounded measurable subsets of g

estimating isogenies on elliptic curves the object of this paper is to prove a new type of estimate for isogenies between elliptic curves this has several diophantine applications effective versions of serre s galois irreducibility theorem and shafarevich s theorem for example which are presented in another paper mw3 later articles will deal with the corresponding problems for abelian varieties of arbitrary dimension right at the beginning we emphasize that we are identifying elliptic curves e with weierstrass equations

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

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

locally optimal knots and tension parameters for exponential splines tension can be applied to cubic splines in order to avoid undesired spurious oscillations this leads to the well known exponential spline in tension it is crucial but unfortunately difficult to find suitable tension parameters of interpolating splines in tension instead of heuristics we propose a simultaneous knot placing and tension setting algorithm for least squares splines in tension which includes interpolating splines in tension as a special case moreover the splines presented here are the foundation of exponential surface splines on fairly arbitrary meshes k o riedel two dimensional splines on fairly arbitrary meshes zamm z angew math mech 85 3 2005 176 188

an explicit theory of heights we consider the problem of explicitly determining the naive height constants for jacobians of hyperelliptic curves for genus 1 it is impractical to apply hilbert s nullstellensatz directly to the defining equations of the duplication law we indicate how this technical difficulty can be overcome by use of isogenies the height constants are computed in detail for the jacobian of an arbitrary curve of genus 2 and we apply the technique to compute generators of j q the mordell weil group for a selection of rank 1 examples

a method for computing the tension parameters in convexity preserving spline in tension interpolation this paper is concerned with the problem of convexity preservng orc preserving interpolation by using exponential splines in tension or est s for this purpose the notion of ac preserving interpolant which is usually employed in spline in tension interpolation is refined and the existence ofc preserving est s is established for the so calledc admissible data sets the problem of constructing ac preserving and visually pleasing est is then treated by combining a generalized newton raphson method due to ben israel with a step length technique which serves the need for visual pleasantness the numerical performance of the so formed iterative scheme is discussed for several examples

zero estimates on group varieties ii the aim of this article is to give an account of a class of estimates that have in recent years been found useful in the theory of transcendental numbers as our space is limited and the estimates are quite technical in nature wc can provide no hints of proofs and we prefer to illustrate the results with examples rather than supply full statements we also mention some of the applications one basic method in transcendence theory can be described in general terms as follows suppose it is desired to establish a certain proposition usually it will be possible to associate with the problem a collection of meromorphic functions ft fnm such a way that assuming the falsity of the proposition imposes strong arithmetical conditions on the functions these enable us to construct a polynomial p p x1 xn such that the function f p f1 fn is not identically zero but has a large number of zeroes by means of analytic interpolation arguments it can then be deduced that takes many small values and by using once more the arithmetical conditions we conclude that has even more zeroes than initially arranged but often merely knowing upper bounds for the degree of p leads to upper bounds for the number of zeroes of and if we are lucky these yield the contradiction that proves the original proposition in this article we shall limit attention to the last stage of this argument as it is here that the most interesting recent progress has taken place a simple example will suffice to illustrate the sort of estimates involved to show that w and e cannot both be algebraic for w 0 suppose they are then we end up with a non zero polynomial p p x y of degree at most l 1 in x and of degree at most jf 1 in y such that the function

residue calculus and effective nullstellensatz multivariate residue calculus in the spirit of j lipman is developed from the computational point of view for example with several variants of the classical transformation law and used in order to make totally explicit the bezout identity and therefore the algebraic nullstellensatz in k x 1 x n where k is an infinite field of arbitrary characteristic such identities provide sharp size estimates for the denominator and the divisors in the bezout identity when k is the quotient field of a factorial regular ring a equipped with a size such as z or f p 1 q the estimates obtained by the authors in a previous work in the case a z are sharpened here while analytic techniques are replaced with an algebraic approach

homogeneous quantization and multiplicities of group representations let b be a compact manifold a cone over b is a principal r bundle x with base b let a x a x be the mapping associated with the action of a r on x x is called a symplectic cone if it possesses a symplectic form such that a a a compact lie group g is said to act in a homogeneous fashion on x if it acts on x in such a way that both and the principal bundle structure are preserved it is known that to such an action one can associate in a fairly canonical way a representation of g on a hilbert space h see 3 in this paper we propose a symplectic recipe for the multiplicities with which h decomposes into g irreducibles and show that this recipe is correct generically

the hamiltonian structure of the maxwell vlasov equations morrison 25 has observed that the maxwell vlasov and poisson vlasov equations for a collisionless plasma can be written in hamiltonian form relative to a certain poisson bracket we derive another poisson structure for these equations by using general methods of symplectic geometry the main ingredients in our construction are the symplectic structure on the co adjoint orbits for the group of canonical transformations and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory our poisson bracket satisfies the jacobi identity whereas morrison s does not 37 our construction also shows where canonical variables can be found and can be applied to the yang mills vlasov equations and to electromagnetic fluid dynamics

circuit complexity proof complexity and polynomial identity testing the ideal proof system we introduce a new and natural algebraic proof system whose complexity measure is essentially the algebraic circuit size of nullstellensatz certificates this enables us to exhibit close connections between effective nullstellensatze proof complexity and algebraic circuit complexity in particular we show that any super polynomial lower bound on any boolean tautology in our proof system implies that the permanent does not have polynomial size algebraic circuits vnp vp we also show that super polynomial lower bounds on the number of lines in polynomial calculus proofs imply the permanent versus determinant conjecture note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound our proof system helps clarify the relationships between previous algebraic proof systems in doing so we highlight the importance of polynomial identity testing pit in proof complexity in particular we use pit to illuminate ac0 p frege lower bounds which have been open for nearly 30 years with no satisfactory explanation as to their apparent difficulty finally we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity using the algebraic structure of our proof system we propose a novel route to such lower bounds although such lower bounds remain elusive this proposal should be contrasted with the difficulty of extending ac0 p circuit lower bounds to ac0 p frege lower bounds

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

the differentiable structure of three remarkable diffeomorphism groups the goal of this paper is to find the lie groups for three well known lie algebras the globally hamiltonian vector fields the infinitesimal quantomorphisms and the homogeneous real valued functions of degree one on the cotangent bundle minus the zero section in all our considerations the underlying manifold is assumed to be compact without boundary before explaining and motivating our results a very brief review of the lie group structure of diffeomorphism groups is in order 9 10 18 if m n are compact boundaryless finite dimensional manifolds rn tn n the tangent bundle of n an hs map s dimm 2 from m to n has by definition all derivatives of order dim m 2 then the space ip m n of all such maps is a hilbert manifold whose tangent space at every point is the hilbert space t i h m n gelid m tn i n g f if e m is a vector bundle the same construction works for all hs sections hs e of e s dim m 2 if n ir hs m n will be denoted by c m ir let i m denote the diffeomorphisms of m of sobolev class h s l i e r t l m if and only if t is bijective and t t l m m are of class h 1 s l m is a topological group and since it is open in hs i m m it is also a hilbert manifold right multiplication r m i m r o t is c for each t y m and if t e s k l m left multiplication l t m i m l t o is c k the inversion map t t 1 in t m is only continuous the tangent maps of r and l at e the identity of a m are given by tr tel x t ox where x yys i m hs i tm the set of all h l vector fields on m the tangent space at e tr coincides with y m which is a sobolev space the bracket x y of x y f s i m is however only of class h one derivative is lost the usual bracket of vector fields is the lie algebra bracket of 5f x m i e if j

on the local global conjecture for integral apollonian gaskets we prove that a set of density one satisfies the local global conjecture for integral apollonian gaskets that is for a fixed integral primitive apollonian gasket almost every in the sense of density admissible passing local obstructions integer is the curvature of some circle in the gasket

k area hofer metric and geometry of conjugacy classes in lie groups given a closed symplectic manifold m we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group ham m by means of the hofer metric on ham m we use pseudo holomorphic curves involved in the definition of the multiplicative structure on the floer cohomology of a symplectic manifold m to estimate this quantity in terms of actions of some periodic orbits of related hamiltonian flows as a corollary we get a new way to obtain agnihotri belkale woodward inequalities for eigenvalues of products of unitary matrices as another corollary we get a new proof of the geodesic property with respect to the hofer metric of hamiltonian flows generated by certain autonomous hamiltonians our main technical tool is k area defined for hamiltonian fibrations over a surface with boundary in the spirit of l polterovich s work on hamiltonian fibrations over s2

vanishing distance phenomena and the geometric approach to sqg in this article we study the induced geodesic distance of fractional order sobolev metrics on the groups of volume preserving diffeomorphisms and symplectomorphisms the interest in these geometries is fueled by the observation that they allow for a geometric interpretation for prominent partial differential equations in the field of fluid dynamics these include in particular the modified constantin lax majda and surface quasi geostrophic equations the main result of this article shows that both of these equations stem from a riemannian metric with vanishing geodesic distance

a simple rational spline and its application to monotonic interpolation to monotonic data we shall consider a class of simple rational splines and their application to monotonic interpolation to monotonic data our method is situated between interpolation with the usual cubic splines and with monotone quadratic splines a selection of numerical results is presented in figs 4 11

shape preserving approximations by polynomials and splines a review of shape preserving approximation methods and algorithms for approxi mating univariate functions or discrete data is given the notion of shape refers to the geometrical behavior of a function s or approximant s graph and usually includes positivity monotonicity and or convexity but in the recent literature the broader concept of shape also includes symmetry gen eralized convexity unimodality lipschitz property possessing peaks or discontinuities etc special stress is put on shape preserving interpolation methods by polynomials and splines of course this text has no pretensions to be complete geywords shape preserving approximation approximation of univariate functions approxima tion of discrete data shape preserving interpolation polynomial spline positivity monotonicity convexity generalized convexity unimodality lipschitz property

a spectral gap theorem in simple lie groups we establish the spectral gap property for dense subgroups generated by algebraic elements in any compact simple lie group generalizing earlier results of bourgain and gamburd for unitary groups

computing heights on elliptic curves we describe how to compute the canonical height of points orn elliptic curves tate has given a rapidly converging series for archimedean local heights over r we describe a modified version of tate s series which also converges over c and give an efficient procedure for calculating local heights at non archimedean places in this way we can calculate heights over number fields having complex embeddings we also give explicit estimates for the tail of our series and present several examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 alxy a3y x3 a2x2 a4x a6 the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 11 viii section 9 or 6 chapter vi the canonical height is an extremely important theoretical tool in the arithmetic theory of elliptic curves being used for such diverse purposes as studying values of l functions 5 numbers of integral points 12 and transcendence theory 9 it is also important as a computational tool such as its use in zagier s algorithm for finding integral points up to large bounds 18 it is thus of interest to have an efficient method for calculating the canonical height of a point the usual definition of h as a limit h p limn 0 4 nh x 2np is not practical for computation instead one uses the fact that the canonical height can be written as a sum of local heights one term for each distinct absolute value on k 2 h p e na p vemk for example if k q then mk can be identified with the set of rational primes together with the usual absolute value on q the multiplicities nv are chosen so that the product formula holds and so that h is independent of the choice of the field k the local height corresponding to a non archimedean absolute value is given by intersection theory in a well known manner see e g 2 4 or 7 chapter 11 section 5 we will describe a quick way to compute non archimedean local heights in section 5 the local height for an archimedean absolute value is given by a transcendental function and so efficient computation is somewhat more difficult j tate 15 1988 american mathematical society 0025 5718 88 1 00 25 per page 339 received august 20 1987 revised october 21 1987 1980 mathematics subject classification 1985 revision primary 11g05 14k07 11d25 this work was partially supported by nsf grant dms 8612393 current address mathematics department brown university providence ri 02912 this content downloaded from 207 46 13 103 on thu 20 oct 2016 04 37 48 utc all use subject to http about jstor org terms 340 joseph h silverman has given an easily computed power series which works for real absolute values precisely for a given curve e and point p x y he gives a sequence of easily computed numbers co cl so that

small degree solutions for the polynomial bezout equation abstract we give an overview of the ideas from complex analysis which provide the solution q j of the polynomial bezout equation p 1 q 1 p m q m 1 with the essentially smallest possible degree

singular cosphere bundle reduction this paper studies singular contact reduction for cosphere bundles at the zero value of the momentum map a stratification of the singular quotient finer than the contact one and better adapted to the bundle structure of the problem is obtained the strata of this new stratification are a collection of cosphere bundles and coisotropic or legendrian submanifolds of their corresponding contact components

sharp estimates for the arithmetic nullstellensatz we present sharp estimates for the degree and the height of the polynomials in the nullstellensatz over the integer ring zz the result improves previous work of philippon berenstein yger and krick pardo we also present degree and height estimates of intrinsic type which depend mainly on the degree and the height of the input polynomial system as an application we derive an effective arithmetic nullstellensatz for sparse polynomial systems the proof of these results relies heavily on the notion of local height of an affine variety defined over a number field we introduce this notion and study its basic properties

a convexity preservingc2 parametric rational cubic interpolation ac 2 parametric rational cubic interpolantr t x t i y t j t t 1 t n to data s xj yj j 1 n is defined in terms of non negative tension parameters j j 1 n 1 letp be the polygonal line defined by the directed line segments joining the points x j y j t 1 n sufficient conditions are derived which ensure thatr t is a strictly convex function on strictly left right winding polygonal line segmentsp it is then proved that there always exist j j 1 n 1 for whichr t preserves the local left righ winding properties of any polygonal linep an example application is discussed

effective bezout identities in q z1 zn abstract if p sub 1 p sub m are n variate polynomials with integral coefficients and no common zeros in c exp n brownawell has shown in 1986 that there exist q sub 1 q sub m polynomials with integral coefficients and nu is an element of z such that p sub 1 q sub 1 p sub m q sub m nu and max deg q sub j max deg p sub j exp n on the other hand if h logarithm of the largest coefficient of all the p sub j and h sub 1 is the corresponding quantity for the q sub j then there is no sharp estimate of h sub 1 in terms of h and max deg p sub j in this paper we show that when the variety of common zeros at infinity of the p sub j is discrete then essentially we have h sub 1 d exp cn h for an absolute constant c if there were an algorithm to compute the q sub j in d exp cn time one would obtain exactly the above estimate current algorithms require about d exp n squared operations

ideals generated by quadrics exhibiting double exponential degrees abstract we modify the example of mayr and meyer to give an example of a polynomial ideal generated by quadrics which exhibits double exponential growth for the ideal membership problem we also show that the homogenization of this example gives a set of elements which has a minimal syzygy of double exponential degree

specializations of finitely generated subgroups of abelian varieties given a generic mordell weil group over a function field we can specialize it down to a number field it has been known for some time that the resulting homomorphism of groups is injective infinitely often we prove that this is in fact true almost always in a sense that is quantitatively nearly best possible