Rational Gauss-Chebyshev quadrature formulas for complex poles outside

In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside to arbitrary complex poles outside . The zeros of these orthogonal rational functions are not necessarily real anymore. By using the related para-orthogonal functions, however, we obtain an expression for the nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside .

     

On computing rational Gauss-Chebyshev quadrature formulas

We provide an algorithm to compute the nodes and weights for Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside . Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order . This method is based on the derivation of explicit expressions for Chebyshev orthogonal rational functions, which are (thus far) the only examples of explicitly known orthogonal rational functions on with arbitrary real poles outside this interval.

     

Computing the Ehrhart quasi-polynomial of a rational simplex

We present a polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasi-polynomial of a rational simplex. Previously such algorithms were known for integer simplices and for rational polytopes of a fixed dimension. The algorithm is based on the formula relating the th coefficient of the Ehrhart quasi-polynomial of a rational polytope to volumes of sections of the polytope by affine lattice subspaces parallel to -dimensional faces of the polytope. We discuss possible extensions and open questions.

     

Efficient solution of rational conics

We present efficient algorithms for solving Legendre equations over (equivalently, for finding rational points on rational conics) and parametrizing all solutions. Unlike existing algorithms, no integer factorization is required, provided that the prime factors of the discriminant are known.     

Primes generated by elliptic curves

For a rational elliptic curve in Weierstrass form, Chudnovsky and Chudnovsky considered the likelihood that the denominators of the -coordinates of the multiples of a rational point are squares of primes. Assuming the point is the image of a rational point under an isogeny, we use Siegel's Theorem to prove that only finitely many primes will arise. The same question is considered for elliptic curves in homogeneous form, prompting a visit to Ramanujan's famous taxi-cab equation. Finiteness is provable for these curves with no extra assumptions. Finally, consideration is given to the possibilities for prime generation in higher rank.

     

Height Uniformity for Algebraic Points on Curves

We recall the main result of L. Caporaso, J. Harris, and B. Mazur's 1997 paper of Uniformity of rational points. It says that the Lang conjecture on the distribution of rational points on varieties of general type implies the uniformity for the numbers of rational points on curves of genus at least 2. In this paper we will investigate its analogue for their heights under the assumption of the Vojta conjecture. Basically, we will show that the Vojta conjecture gives a naturally expected simple uniformity for their heights.     

Sieving for rational points on hyperelliptic curves

We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even when our curve has points over all localizations . We illustrate the practicality of the method with some examples of hyperelliptic curves of genus .

     

Multivariate rational data fitting: general data structure,maximal accuracy and object orientation

Sections 1 and 2 discuss the advantages of an object-oriented implementation combined with higher floating-point arithmetic, of the algorithms available for multivariate data fitting using rational functions. Section 1 will in particular explain what we mean by higher arithmetic. Section 2 will concentrate on the concepts of object orientation. In sections 3 and 4 we shall describe the generality of the data structure that can be dealt with: due to some new results virtually every data set is acceptable right now, with possible coalescence of coordinates or points. In order to solve the multivariate rational interpolation problem the data sets are fed to different algorithms depending on the structure of the interpolation points in then-variate space.This text is a preparatory publication for the development of a scientific expert system for multivariate rational interpolation. The issues addressed are relevant to the implementation of such a system.     

Short rational generating functions for lattice point problems

We prove that for any fixed the generating function of the projection of the set of integer points in a rational -dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert bases of rational cones, have short rational generating functions provided certain parameters (the dimension and the number of generators) are fixed. It follows then that many computational problems for such sets (for example, finding the number of positive integers not representable as a non-negative integer combination of given coprime positive integers ) admit polynomial time algorithms. We also discuss a related problem of computing the Hilbert series of a ring generated by monomials.

     

Rational isomorphisms between K-theories and cohomology theories

The well known isomorphism relating the rational algebraic K-theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the Segre map) of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced that establishes a useful general criterion for a natural transformation of functors on quasi-projective complex varieties to induce a homotopy equivalence of semi-topological singular complexes. Since semi-topological K-theory and morphic cohomology can be formulated as the semi-topological singular complexes associated to algebraic K-theory and motivic cohomology, this criterion provides a rational isomorphism between the semi-topological K-theory groups and the morphic cohomology groups of a smooth complex variety. Consequences include a Riemann-Roch theorem for the Chern character on semi-topological K-theory and an interpretation of the topological filtration on singular cohomology groups in K-theoretic terms.     

On the -invariant of rational surface singularities

We show that for rational surface singularities with odd determinant the -invariant defined by W. Neumann is an obstruction for the link of the singularity to bound a rational homology 4-ball. We identify the -invariant with the corresponding correction term in Heegaard Floer theory.

     

A rational Arnoldi process with applications

The rational Arnoldi process is a popular method for the computation of a few eigenvalues of a large non‐Hermitian matrix and for the approximation of matrix functions. The method is particularly attractive when the rational functions that determine the process have only few distinct poles , because then few factorizations of matrices of the form A ? z_jI have to be computed. We discuss recursion relations for orthogonal bases of rational Krylov subspaces determined by rational functions with few distinct poles. These recursion formulas yield a new implementation of the rational Arnoldi process. Applications of the rational Arnoldi process to the approximation of matrix functions as well as to the computation of eigenvalues and pseudospectra of A are described. The new implementation is compared to several available implementations.     

A parallel algorithm for discrete least squares rational approximation

Summary A new method for discrete least squares linearized rational approximation is presented. It generalizes the algorithm of Rutishauser-Gragg-Harrod-Reichel for discrete least squares polynomial approximation to the rational case. The algorithm is fast in the sense that it requires orderm computation time wherem is the number of data points and is the degree of the approximant. We describe how this algorithm can be implemented in parallel.     

Rational -equivariant homotopy theory

We give an algebraicization of rational -equivariant homotopy theory. There is an algebraic category of `` -systems' which is equivalent to the homotopy category of rational -simply connected -spaces. There is also a theory of ``minimal models' for -systems, analogous to Sullivan's minimal algebras. Each -space has an associated minimal -system which encodes all of its rational homotopy information, including its rational equivariant cohomology and Postnikov decomposition.

     

Primitive points on constant elliptic curves over function fields

Letf:CE be a non-constant rational map between curves over a finite field, whereE is elliptic. We estimate the number of rational points ofC whose image underf generate the group of rational points ofE.     

Generalized pseudoinverses of matrix valued functions

We define the pseudoinverse (resp. a generalized pseudoinverse) of a matrix-valued functionF to be the functionF ^x such that, for each in the domain ofF, F ^x () is the inverse (resp. a generalized inverse) of the matrixF(). We derive a state space formula for a generalized pseudoinverse of a rational matrix function without a pole or zero at infinity. This derivation makes use of the theorem characterizing the factorization of a nonregular rational matrix functionW in terms of the decomposition of the state space of a realization ofW. We also give a formula for a generalized pseudoinverse of an arbitrary rational matrix function in the form of a centered realization. We indicate some applications of generalized pseudoinverses of matrix valued functions.