Space Of Rational Functions Research Articles

An inner product on Adelic measures with applications to the Arakelov–Zhang pairing

We define an inner product on a vector space of adelic measures over a number field. We find that the norm induced by this inner product governs weak convergence at each place of K K . The canonical adelic measure associated to a rational map is in this vector space, and the square of the norm of the difference of two such adelic measures is the Arakelov–Zhang pairing from arithmetic dynamics. We find that the norm of a canonical adelic measure associated to a rational map is commensurate with a height on the space of rational functions with fixed degree. As a consequence, we show that the Arakelov–Zhang pairing of two rational maps f f and g g is, when holding g g fixed, commensurate with the height of f f .

Parabolic bifurcation loci in the spaces of rational functions

We give a geometric description of the parabolic bifurcation locus in the space Rat d of all rational functions on of degree d > 1, generalising the study by Morton and Vivaldi in the case of monic polynomials. The results are new even for quadratic rational functions.

Regular polynomial automorphisms in the space of planar quadratic rational maps

In this paper, we describe the semistable quotient of the set of regular polynomial automorphisms $${\mathcal {H}}_2^2$$ in the semistable locus of the moduli space of quadratic rational maps, using the portrait moduli space of rational maps with a fixed point. We also provide the parametrization of $${\mathcal {H}}_2^2$$ using two invariants, $$\det f$$ and $${\text {tr}}\,f$$ .

Semi-Hyperbolic Maps Are Rare

We prove in this article that the set of semi-hyperbolic rational maps has Lebesgue measure zero in the space of rational maps on the Riemann sphere for a fixed degree d = 2. This extends the earlier result [3] to its full generality.

Inequalities pertaining to rational functions with prescribed poles

Let Rn be the space of rational functions with prescribed poles. If t1, t2, ..., tn are the zeros of B(z) + ? and s1, s2, ..., sn are zeros of B(z) - ?, where B(z) is the Blaschke product and ? ? T; then for z ? T |r'(z)| ? |B'(z)|/2 [(max 1?k?n |r(tk)|)2 + (max 1 ? k ? n |r(sk)|)2]. Let r, s ? Rn and assume s has all its n zeros in D- ? T and |r(z)| ? |s(z)| for z ? , then for any ? with |?| ? 1/2 and for z ? T |r'(z) + ?B'(z)r(z)|? |s'(z) + ?B'(z)s(z)|. In this paper, we consider a more general class of rational functions rof ? Rm*n, defined by (rof)(z) = r(f(z)), where f(z) is a polynomial of degree m* and prove some generalizations of the above inequalities.

Matrix methods for quadrature formulas on the unit circle. A survey

In this paper we give a survey of some results concerning the computation of quadrature formulas on the unit circle.Like nodes and weights of Gauss quadrature formulas (for the estimation of integrals with respect to measures on the real line) can be computed from the eigenvalue decomposition of the Jacobi matrix, Szegő quadrature formulas (for the approximation of integrals with respect to measures on the unit circle) can be obtained from certain unitary five-diagonal or unitary Hessenberg matrices that characterize the recurrence for an orthogonal (Laurent) polynomial basis. These quadratures are exact in a maximal space of Laurent polynomials.Orthogonal polynomials are a particular case of orthogonal rational functions with prescribed poles. More general Szegő quadrature formulas can be obtained that are exact in certain spaces of rational functions. In this context, the nodes and the weights of these rules are computed from the eigenvalue decomposition of an operator Möbius transform of the same five-diagonal or Hessenberg matrices.

The Collet–Eckmann condition for rational functions on the Riemann sphere

We show that the set of Collet–Eckmann maps has positive Lebesgue measure in the space of rational maps on the Riemann sphere for any fixed degree d ≥ 2.

Computing rational Gauss–Chebyshev quadrature formulas with complex poles: The algorithm

We provide an algorithm to compute arbitrarily many nodes and weights for rational Gauss–Chebyshev quadrature formulas integrating exactly in spaces of rational functions with complex poles outside [−1, 1]. 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 O ( n ) . This algorithm is based on the derivation of explicit expressions for the Chebyshev (para-)orthogonal rational functions on [−1, 1] with arbitrary complex poles outside this interval.

Rational Gauss-Chebyshev quadrature formulas for complex poles outside $[-1,1

In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside [-1,1] to arbitrary complex poles outside [-1,1]. 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 [-1,1].

Structured eigenvalue problems for rational gauss quadrature

The connection between Gauss quadrature rules and the algebraic eigenvalue problem for a Jacobi matrix was first exploited in the now classical paper by Golub and Welsch (Math. Comput. 23(106), 221–230, 1969). From then on many computational problems arising in the construction of (polynomial) Gauss quadrature formulas have been reduced to solving direct and inverse eigenvalue problems for symmetric tridiagonals. Over the last few years (rational) generalizations of the classical Gauss quadrature formulas have been studied, i.e., formulas integrating exactly in spaces of rational functions. This paper wants to illustrate that stable and efficient procedures based on structured numerical linear algebra techniques can also be devised for the solution of the eigenvalue problems arising in the field of rational Gauss quadrature.

Moduli spaces with external fields

We consider the geometric structures on the moduli space of static finite energy solutions to the 2 + 1 -dimensional unitary chiral model with the Wess–Zummino–Witten (WZW) term. It is shown that the magnetic field induced by the WZW term vanishes when restricted to the moduli spaces constructed from the Grassmannian embeddings, so that the slowly moving solitons can in some cases be approximated by a geodesic motion on a space of rational maps from C P 1 to the Grassmannian.

Counting Ramified Coverings and Intersection Theory on Hurwitz Spaces II (Local Structure of Hurwitz Spaces Combinatorial Results)

The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these numbers belong to a very particular subalgebra of the algebra of power series.

Fixed points of maps on the space of rational functions

Given integers s, t, define a function φs,t on the space of all formal series expansions by φs,t( P anx ) = P asn+tx . For each function φs,t, we determine the collection of all rational functions whose Taylor expansions at zero are fixed by φs,t. This collection can be described as a subspace of rational functions whose basis elements correspond to certain s-cyclotomic cosets associated with the pair (s, t). Subject Class: Primary 33

A Map on the Space of Rational Functions

We describe dynamical properties of a map F defined on the space of rational functions. The fixed points of F are classified and the long time behavior of a subclass is described in terms of Eulerian polynomials.

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 [ − 1 , 1 ] [-1,1] . 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 O ( n ) O(n) . 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 [ − 1 , 1 ] [-1,1] with arbitrary real poles outside this interval.

Orthogonality of some sequences of the rational functions and the Müntz polynomials

In this paper we investigate the inner products in the space of rational functions and the space of Müntz polynomials. We prove that the orthogonality of a sequence in one of mentioned spaces can be induced by the orthogonality of the corresponding sequence in another space. Finally, we point to several special cases, i.e., some very different classes of well–known functions we represent on a unique way.

Group Actions on Spaces of Rational Functions

Let \mathrm{Hol}_d be the space consisting of all holomorphic maps f : S^2 → S^2 of degree d . The group \mathrm{Hol}_1 = \mathrm{PSL}_2(ℂ) acts on \mathrm{Hol}_d freely by the post-composition and we shall study the orbit space X_d = \mathrm{Hol}_1\backslash \mathrm{Hol}_d . As an application, we shall determine the homotopy types of the universal covering spaces of \mathrm{Hol}_d and X_d explicitly.

The Atiyah–Hitchin bracket and the open Toda lattice

The dynamics of the finite nonperiodic Toda lattice is an isospectral deformation of the finite three-diagonal Jacobi matrix. It is known since the work of Stieltjes that such matrices are in one-to-one correspondence with their Weyl functions. These are rational functions mapping the upper half-plane into itself. We consider representations of the Weyl functions as a quotient of two polynomials and exponential representation. We establish a connection between these representations and recently developed algebraic-geometrical approach to the inverse problem for Jacobi matrix. The space of rational functions has natural Poisson structure discovered by Atiyah and Hitchin. We show that an invariance of the AH structure under linear-fractional transformations leads to two systems of canonical coordinates and two families of commuting Hamiltonians. We establish a relation of one of these systems with Jacobi elliptic coordinates.

Dependent rational points on curves over finite fields - Lefschetz theorems and exponential sums

For an algebraic curve defined over Fq we study the probability that t randomly chosen Fq-rational points on the curve impose dependent conditions on the functions in a given τ-dimensional vector space of rational functions on the curve. This probability tends to be close to 1q. The results have applications in the assessment of the performance of decoding algorithms for algebraic geometry codes. The proofs involve a geometric construction. Lefschetz theorem for quasi-projective varieties and majorizations of exponential sums.

Poisson brackets on rational functions and multi-Hamiltonian structure for integrable lattices

We introduce a family of compatible Poisson brackets on the space of rational functions with denominator of a fixed degree and use it to derive a multi-Hamiltonian structure for a family of integrable lattice equations that includes both the standard and the relativistic Toda lattices.