Field Of Meromorphic Functions Research Articles

On special values of generators of modular function fields

Let N be a positive integer and ζN be a primitive Nth root of unity. Let X(N) be the modular curve of level N and FX(N),Q(ζN) be its field of meromorphic functions with Fourier coefficients in Q(ζN). Let u and v be nonconstant functions in FX(N),Q(ζN) which generate a sufficiently large subfield of FX(N),Q(ζN) over Q(ζN). In this paper, we give sufficient conditions for generating the ray class field of an imaginary quadratic field K modulo N over K(ζN) by the special values of u and v.

Hypertranscendency of perturbations of hypertranscendental functions

Inspired by the work of Bank on the hypertranscendence of Γeh where Γ is the Euler gamma function and h is an entire function, we investigate when a meromorphic function feg cannot satisfy any algebraic differential equation over certain field of meromorphic functions, where f and g are meromorphic and entire on the complex plane, respectively. Our results (Theorem 1 and 2) give partial solutions to Bank's Conjecture (1977) on the hypertranscendence of Γeh. We also give some sufficient conditions for hypertranscendence of meromorphic function of the form f+g, f⋅g and f∘g in Theorem 3 and 4.

Meromorphic function fields closed by partial derivatives

We characterize meromorphic function fields closed by partial derivatives in n variables.

Construction of Two Parametric Deformation of KdV-Hierarchy and Solution in Terms of Meromorphic Functions on the Sigma Divisor of a Hyperelliptic Curve of Genus 3

Buchstaber and Mikhailov introduced the polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. In our previous paper, we constructed the field of meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3 and solutions of the systems for $g=3$ by these functions. In this paper, as an application of our previous results, we construct two parametric deformation of the KdV-hierarchy. This new system is integrated in the meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3. In Section 8 of our previous paper [Funct. Anal. Appl. 51 (2017), 162-176], there are miscalculations. In appendix of this paper, we correct the errors.

Independence of Artin L-functions

Abstract Let K / ℚ {K/\mathbb{Q}} be a finite Galois extension. Let χ 1 , … , χ r {\chi_{1},\ldots,\chi_{r}} be r ≥ 1 {r\geq 1} distinct characters of the Galois group with the associated Artin L-functions L ⁢ ( s , χ 1 ) , … , L ⁢ ( s , χ r ) {L(s,\chi_{1}),\ldots,L(s,\chi_{r})} . Let m ≥ 0 {m\geq 0} . We prove that the derivatives L ( k ) ⁢ ( s , χ j ) {L^{(k)}(s,\chi_{j})} , 1 ≤ j ≤ r {1\leq j\leq r} , 0 ≤ k ≤ m {0\leq k\leq m} , are linearly independent over the field of meromorphic functions of order < 1 {<1} . From this it follows that the L-functions corresponding to the irreducible characters are algebraically independent over the field of meromorphic functions of order < 1 {<1} .

State feedback linearization of nonlinear control systems on homogeneous time scales

The paper addresses the state feedback linearization problem for nonlinear systems, defined on homogeneous time scale. Necessary and sufficient solvability conditions are given within the algebraic framework of differential one-forms. The conditions concerning the exact dynamic state feedback linearization are equivalent to the property of differential flatness of the system. An output function which defines a right invertible system without zero-dynamics is shown to exist if and only if the basis of some space of one-forms can be transformed, via polynomial matrix operator over the field of meromorphic functions, into a system of exact one-forms. The results extend the corresponding results for the continuous-time case.

Cartan’s conjecture for moving hypersurfaces

Let f be a holomorphic curve in $${\mathbb {P}}^n({{\mathbb {C}}})$$ and let $$\mathcal {D}=\{D_1,\ldots ,D_q\}$$ be a family of moving hypersurfaces defined by a set of homogeneous polynomials $$\mathcal {Q}=\{Q_1,\ldots ,Q_q\}$$ . For $$j=1,\ldots ,q$$ , denote by $$Q_j=\sum \nolimits _{i_0+\cdots +i_n=d_j}a_{j,I}(z)x_0^{i_0}\cdots x_n^{i_n}$$ , where $$I=(i_0,\ldots ,i_n)\in {\mathbb {Z}}_{\ge 0}^{n+1}$$ and $$a_{j,I}(z)$$ are entire functions on $${{\mathbb {C}}}$$ without common zeros. Let $$\mathcal {K}_{\mathcal {Q}}$$ be the smallest subfield of meromorphic function field $$\mathcal {M}$$ which contains $${{\mathbb {C}}}$$ and all $$\frac{a_{j,I'}(z)}{a_{j,I''}(z)}$$ with $$a_{j,I''}(z)\not \equiv 0$$ , $$1\le j\le q$$ . In previous known second main theorems for f and $$\mathcal {D}$$ , f is usually assumed to be algebraically nondegenerate over $$\mathcal {K}_{\mathcal {Q}}$$ . In this paper, we prove a second main theorem in which f is only assumed to be nonconstant. This result can be regarded as a generalization of Cartan’s conjecture for moving hypersurfaces.

Algebraic dimension of complex nilmanifolds

Let M be a complex nilmanifold, that is, a compact quotient of a nilpotent Lie group endowed with an invariant complex structure by a discrete lattice. A holomorphic differential on M is a closed, holomorphic 1-form. We show that a(M)⩽k, where a(M) is the algebraic dimension a(M) (i.e. the transcendence degree of the field of meromorphic functions) and k is the dimension of the space of holomorphic differentials. We prove a similar result about meromorphic maps to Kähler manifolds.

Remarks on the realization of time-varying systems

The realization problem of nonlinear time-varying input–output equations is considered. Differentials of the state coordinates, necessary for realization, are determined by the vector space of differential one-forms, spanned over the field of meromorphic functions. Formulas for computing the basis one-forms are given, based on the Euclidean division of non-commutative polynomials. Moreover, it is shown that in the case of a reducible system, the subspace admits a basis with certain structure, explicitly related to reduced input–output equations.

The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 and applications

The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus $3$ is described in terms of the gradient of it's sigma function. Solutions of corresponding families of polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals are constructed as an application. These systems were introduced in the work of V. M. Buchstaber and A. V. Mikhailov on the basis of commuting vector fields on the symmetric square of algebraic curves.

TRANSCENDENCE OVER MEROMORPHIC FUNCTIONS

In this short note, considering functions, we show that taking an asymptotic viewpoint allows one to prove strong transcendence statements in many general situations. In particular, as a consequence of a more general result, we show that if$F(z)\in \mathbb{C}[[z]]$is a power series with coefficients from a finite set, then$F(z)$is either rational or it is transcendental over the field of meromorphic functions.

NORMAL BASES FOR MODULAR FUNCTION FIELDS

We provide a concrete example of a normal basis for a finite Galois extension which is not abelian. More precisely, let$\mathbb{C}(X(N))$be the field of meromorphic functions on the modular curve$X(N)$of level$N$. We construct a completely free element in the extension$\mathbb{C}(X(N))/\mathbb{C}(X(1))$by means of Siegel functions.

Definability of Frobenius orbits and a result on rational distance sets

We prove that the first order theory of (possibly transcendental) meromorphic functions of positive characteristic \(p>2\) is undecidable. We also establish a negative solution to an analogue of Hilbert’s tenth problem for such fields of meromorphic functions, for Diophantine equations including vanishing conditions. These undecidability results are proved by showing that the binary relation \(\exists s\ge 0, f=g^{p^s}\) is positive existentially definable in such fields. We also prove that the abc conjecture implies a solution to the Erdos–Ulam problem on rational distance sets. These two seemingly distant topics are addressed by a study of power values of bivariate polynomials of the form F(X)G(Y).

Symbolic computation of Drazin inverses by specializations

In this paper, we show how to reduce the computation of Drazin inverses over certain computable fields to the computation of Drazin inverses of matrices with rational functions as entries. As a consequence we derive a symbolic algorithm to compute the Drazin inverse of matrices whose entries are elements of a finite transcendental field extension of a computable field. The algorithm is applied to matrices over the field of meromorphic functions, in several complex variables, on a connected domain and to matrices over the field of Laurent formal power series.

Some Results on Ordinary Differential Operators with Periodic Coefficients

For a general ordinary differential operator \(\mathcal {L}\) with periodic coefficients we prove that the characteristic polynomial of the Floquet matrix is irreducible over the field of meromorphic functions. We also consider a multipoint eigenvalue problem and show that its eigenspaces are spanned by pure or generalized Floquet solutions. Finally, at the end of the paper we mention some relevant conjectures and open questions.

Galois theory, coverings, and Riemann surfaces

Chapter 1 Galois Theory: 1.1 Action of a Solvable Group and Representability by Radicals.- 1.2 Fixed Points under an Action of a Finite Group and Its Subgroups.- 1.3 Field Automorphisms and Relations between Elements in a Field.- 1.4 Action of a k-Solvable Group and Representability by k-Radicals.- 1.5 Galois Equations.- 1.6 Automorphisms Connected with a Galois Equation.- 1.7 The Fundamental Theorem of Galois Theory.- 1.8 A Criterion for Solvability of Equations by Radicals.- 1.9 A Criterion for Solvability of Equations by k-Radicals.- 1.10 Unsolvability of Complicated Equations by Solving Simpler Equations.- 1.11 Finite Fields.- Chapter 2 Coverings: 2.1 Coverings over Topological Spaces.- 2.2 Completion of Finite Coverings over Punctured Riemann Surfaces.- Chapter 3 Ramified Coverings and Galois Theory: 3.1 Finite Ramified Coverings and Algebraic Extensions of Fields of Meromorphic Functions.- 3.2 Geometry of Galois Theory for Extensions of a Field of Meromorphic Functions.- References.- Index

A principal function problem and the field of meromorphic functions

Let R be an open Riemann surface and A an open subset of R. For a given meromorphic function s on A, we show, under an additional condition on A, that there exists a meromorphic function q on R such that both q/s and s/q are bounded functions on A.

THE SIGMA FUNCTION FOR WEIERSTRASS SEMIGROUPS 〈3, 7, 8〉 AND 〈6, 13, 14, 15, 16〉

Compact Riemann surfaces and their abelian functions are instrumental to solve integrable equations; more recently the representation theory of the Monster and related modular forms have pointed to the relevance of τ-functions, which are in turn connected with a specific type of abelian function, the (Kleinian) σ-function. Klein originally generalized Weierstrass' σ-function to hyperelliptic curves in a geometric way, then from a modular point of view to trigonal curves. Recently a modular generalization for all curves was given, as well as the geometric one for certain affine planar curves, known as (n, s) curves, and their generalizations known as "telescopic" curves. This paper proposes a construction of σ-functions based on the nature of the Weierstrass semigroup at one point of the Riemann surface as a generalization of the construction for plane affine models of the Riemann surface. Our examples are not telescopic. Because our definition is algebraic, in the sense of being related to the field of meromorphic functions on the curve, we are able to consider properties of the σ-functions such as Jacobi inversion formulae, and to observe relationships between their properties and those of a Norton basis for replicable functions, in turn relevant to the Monstrous Moonshine.

Complex structures on product of circle bundles over complex manifolds

Soient L ¯ i →X i des fibrés en droites holomorphes sur des variétés complexes compactes, pour i=1,2. Soit S i le fibré en cercles associé par rapport à un produit scalaire hermitienne sur L ¯ i . On construit des structures complexes sur S=S 1 ×S 2 dites de type scalaire, diagonal, ou linéaire. Bien que des structures de type scalaire existent toujours, on construit des structures plus générales de type diagonal mais non-scalaire dans le cas où les L ¯ i sont des (ℂ * ) n i fibrés équivariants qui vérifient certaines hypothèses supplémentaires. Les structures complexes de type linéaire sont des variétés des drapeaux (généralisées) et les L i sont des fibrés en droites amples négatifs. Lorsque H 1 (X 1 ;ℝ)=0 et c 1 (L ¯ 1 ) est non-nulle la variété compacte S n’admet pas de structure symplectique et donc elle est non-Kählerienne par rapport à toute structure complexe.

Exceptional values of p-adic analytic functions and derivatives

Let K be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value | · |. Given a meromorphic function f in K (resp. inside an ‘open’ disk D) we check that the field of small meromorphic functions in K (resp. inside D) is algebraically closed in the whole field of meromorphic functions in K (resp. inside D). If two analytic functions h, l in K, other than affine functions, satisfy h′l − hl′ = c ∈ K, then c = 0. The space of the entire functions solutions of the equation y″ = φy, with φ a meromorphic function in K or an unbounded meromorphic function in D, is at most of dimension 1. If a meromorphic function in K has no multiple pole, then f′ has no exceptional value. If f is meromorphic with finitely many zeroes then for every c ≠ 0, f′ − c has an infinity of zeroes. If is not a constant or an affine function and if f has no simple pole with a residue equal to 1, then f′ + f 2 admits at least one zero. When K has residue characteristic zero, we extend some results for entire functions to analytic functions in D.