Algebraic Independence Research Articles

Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic

In analogy with the Riemann zeta function at positive integers, for each finite field F_p^r with fixed characteristic p we consider Carlitz zeta values zeta_r(n) at positive integers n. Our theorem asserts that among the zeta values in {zeta_r(1), zeta_r(2), zeta_r(3), ... | r = 1, 2, 3, ...}, all the algebraic relations are those algebraic relations within each individual family {zeta_r(1), zeta_r(2), zeta_r(3), ...}. These are the algebraic relations coming from the Euler-Carlitz relations and the Frobenius relations. To prove this, a motivic method for extracting algebraic independence results from systems of Frobenius difference equations is developed.

Algebraic independence of periods and logarithms of Drinfeld modules

Let ρ \rho be a Drinfeld A A -module with generic characteristic defined over an algebraic function field. We prove that all of the algebraic relations among periods, quasi-periods, and logarithms of algebraic points on ρ \rho are those coming from linear relations induced by endomorphisms of ρ \rho .

APPROXIMATIONS FONCTIONNELLES DES COURBES DES ESPACES PROJECTIFS

Algebraic approximation to points in projective spaces offers a new and more flexible approach to algebraic independence theory. When working over the field of algebraic numbers, it leads to open conjectures in higher dimension extending known results in Diophantine approximation. We show here that over the algebraic closure of a function field in one variable, the analog of these conjectures is true. We also derive transfer lemmas which have applications in the study of multiplicity estimates, for example.

New results on algebraic independence with Mahlerʼs method

We give some new results on algebraic independence in the frame of Mahlerʼs method, including algebraic independence of values at transcendental points. We also give some new measures of algebraic independence. In particular, our results furnish for n ⩾ 1 arbitrarily large new examples of families of members ( θ 1 , … , θ n ) ∈ R n normal in the sense of the definition formulated by G. Chudnovsky (1980) [2].

Algebraic independence of infinite products generated by Fibonacci numbers

The aim of this paper is to establish necessary and sufficient conditions for certain infinite products generated by Fibonacci numbers and by Lucas numbers to be algebraically independent.

Special values of Drinfeld modular forms and algebraic independence

Let A be a polynomial ring in one variable over a finite field and k be its fraction field. Let f be a Drinfeld modular form of nonzero weight for a congruence subgroup of GL2(A) so that the coefficients of the q∞-expansion of f are algebraic over k. We consider n CM points α1, . . . , αn on the Drinfeld upper half plane for which the quadratic fields k(α1), . . . , k(αn) are pairwise distinct. Suppose that f is non-vanishing at these n points. Then we prove that f(α1), . . . , f(αn) are algebraically independent over k.

Algebraic independence of generalized MMM–classes

The generalized Morita-Miller-Mumford classes of a smooth oriented manifold bundle are defined as the image of the characteristic classes of the vertical tangent bundle under the Gysin homomorphism. We show that if the dimension of the manifold is even, then all MMM-classes in rational cohomology are nonzero for some bundle. In odd dimensions, this is also true with one exception: the MMM-class associated with the Hirzebruch $\cL$-class is always zero. We also show a similar result for holomorphic fibre bundles.

A generalisation of Fuhrmann's rank condition for discrete dynamic systems

For discrete-time linear systems, controllability and reachability are not equivalent. Instead of the well-known Kalman’s rank condition, which characterises reachability, controllability to origin of the time invariant, discrete-time linear system is equivalent to the Fuhrmann’s rank condition. In the first part of this paper, we prove that controllability to origin of time varying discrete-time linear systems, under a difference-algebraic condition, is equivalent to a generalised Fuhrmann’s rank condition. In the second part, we prove that reachability and observability for time varying discrete-time linear systems are equivalent to a structured Kalman’s rank condition, under the difference algebraic independence of the structure matrices.

Algebraic independence results for reciprocal sums of Fibonacci numbers

Algebraic independence results for reciprocal sums of Fibonacci numbers

Penetration of sub-micron aerosol droplets in composite cylindrical filtration elements

Advection–diffusion transport of aerosol droplets in composite cylindrical filtration elements is analyzed and compared to experimental data. The penetration, characterizing the fraction of droplets that passes through the pores of a filtration element, is quantified for a range of flow rates. The advection–diffusion transport in a laminar Poiseuille flow is treated numerically for slender pores using a finite difference approach in cylindrical coordinates. The algebraic dependence of the penetration on the Peclet number as predicted theoretically, is confirmed by experimental findings at a variety of aspect ratios of the cylindrical pores. The effective penetration associated with a composite filtration element consisting of a set of parallel cylindrical pores is derived. The overall penetration of heterogeneous composite filtration elements shows an algebraic dependence to the fourth power on the radii of the individual pores that are contained. This gives rise to strong variations in the overall penetration in cases with uneven distributions of pore sizes, highly favoring filtration by the larger pores. The overall penetration is computed for a number of basic geometries, providing a point of reference for filtration design and experimental verification.

Algebraic independence over ℚ p of values of analytic functions at points from ℂ p

General theorems on the algebraic independence over ℚ p of the values of analytic functions at points from ℂ p and their applications to particular examples are presented in the paper.

Parameterized Telescoping Proves Algebraic Independence of Sums

Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Moreover, this result throws new light on the question why, for example, Zeilberger’s algorithm fails to find a recurrence with minimal order.

Algebraic dependences of meromorphic mappings in several complex variables

We give some theorems on algebraic dependence of meromorphic mappings in several complex variables into complex projective spaces.

Constructive D-Module Theory with Singular

We overview numerous algorithms in computational D-module theory together with the theoretical background as well as the implementation in the computer algebra system Singular. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein–Sato polynomials and also algorithms, recovering any kind of Bernstein–Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein–Sato polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We also address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.

Independence measures of arithmetic functions

The notion of algebraic dependence in the ring of arithmetic functions with addition and Dirichlet product is considered. Measures for algebraic independence are derived.

Uniqueness of holomorphic curves into abelian varieties

In this paper, we first give a slight improvement of Yamanoi's truncated second main theorem for holomorphic maps into abelian varieties. We then use the result to study the uniqueness problem for such maps. The results obtained generalize and improve E. M. Schmid's uniqueness theorem for holomorphic maps into elliptic curves. In the last section, we consider algebraic dependence for a finite collection of holomorphic curves into an abelian variety.

The algebraic independence of the sum of divisors functions

Let σj(n)=∑d|ndj be the sum of divisors function, and let I be the identity function. When considering only one input variable n, we show that the set of functions {σi}i=0∞∪{I} is algebraically independent. With two input variables, we give a non-trivial identity involving the sum of divisors function, prove its uniqueness, and use it to prove that any perfect number n must have the form n=rσ(r)/(2r−σ(r)), with some restrictions on r. This generalizes the known forms for both even and odd perfect numbers.

Note on Hermitian Jacobi forms

We compare the spaces of Hermitian Jacobi forms (HJF) of weight $k$ and indices 1, 2 with classical Jacobi forms (JF) of weight $k$ and indices 1, 2, 4. Upper bounds for the order of vanishing of HJF at the origin are obtained. We compute the rank of HJF as a module over elliptic modular forms and prove the algebraic independence of the generators in case of index 1. Some related questions are discussed.

Schlanke Körper (Slim fields)

AbstractWe examine fields in which model theoretic algebraic closure coincides with relative field theoretic algebraic closure. These are perfect fields with nice model theoretic behaviour. For example they are exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay. Classes of examples are perfect PAC fields, model complete large fields and henselian valued fields of characteristic 0.

SMALL VALUE ESTIMATES FOR THE ADDITIVE GROUP

We generalize Gel'fond's criterion for algebraic independence to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of ℂ, instead of just one point (one extension deals with a subgroup of ℂ×).