Algebraic Independence Research Articles

Algebrain Independence Measure of Some Contnued Fractions

In the present paper, we prove the algebraic independence of a finite number of real continued fractions which have partial quotients that increase rapidly. Then, we give an algebraic independence measure of these continued fractions.

On the Algebraic Independence of the Values of Functions Associated with Hypergeometric Functions

Functions that are integrals of products of generalized hypergeometric functions of some kind are considered. The conditions for the representability of these functions as a polynomial in hypergeometric ones are found. The necessary and sufficient conditions for the algebraic independence of such functions are established.

Superconductor-Insulator Transition in a Non-Fermi Liquid.

We present a model of a strongly correlated system with a non-Fermi liquid high temperature phase. Its ground state undergoes an insulator to superconductor quantum phase transition (QPT) as a function of a pairing interaction strength. Both the insulator and the superconductor are originating from the same interaction mechanism. The resistivity in the insulating phase exhibits the activation behavior with the activation energy, which goes to zero at the QPT. This leads to a wide quantum critical regime with an algebraic temperature dependence of the resistivity. Upon raising the temperature in the superconducting phase, the model exhibits a finite temperature phase transition to a Bose metal phase, which separates the superconductor from the non-Fermi liquid metal.

Algebraic Independence of the Values of Power Series and Their Derivatives Generated by Linear Recurrences

Using a descent method, we construct certain power series generated by linear recurrences, each of which possesses the following property: The infinite set consisting of all its values and all the values of its derivatives of any order, at any nonzero algebraic numbers within its domain of existence, is algebraically independent. The main theorems of this paper assert that the power series of the form $\sum_{k=0}^{\infty}z^{e_{k}}$, where $\{e_{k}\}_{k\geq0}$ is a linear recurrence with certain admissible properties, have this property. In particular, Main Theorem 1.16 provides a class of $\{e_{k}\}_{k\geq0}$ which is simpler than ever before.

Algebraic independence results for colored multizeta values in characteristic p

This paper studies multizeta values in positive characteristic, which are function field analogs of the Euler–Riemann multizeta values. The objective of this paper is two-fold. We first define mixed Carlitz motives, which is a counterpart to mixed Tate motives in characteristic zero. After that, we identify special mixed Carlitz motives, and use them to derive algebraic independence properties of colored multizeta values. Our results generalize those in the literature on algebraic relations between multizeta values.

A uniform bound on the multiplicity of Dirichlet eigenvalues in planar annuli

We study the multiplicity of Dirichlet eigenvalues in the planar annulus Ωr={x∈R2:r<|x|<1} when r is an algebraic number. Using the algebraic independence results of the Bessel functions developed by Siegel and Shildlovskii, we prove that all Dirichlet eigenvalues here have multiplicity at most 6.

General algebraic dependence relations

General algebraic dependence relations

Algebraic dependences of meromorphic mappings into a projective space sharing few hyperplanes

Abstract In this article, we will show some algebraic dependence theorems for meromorphic mappings into a projective space sharing few hyperplanes. Our result is an improvement of many previous results in this topic.

A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers

A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers

Topological properties and algebraic independence of sets of prime‐representing constants

Let $(c_k)_{k\in \mathbb{N}}$ be a sequence of positive integers. We investigate the set of $A>1$ such that the integer part of $A^{c_1\cdots c_k}$ is always a prime number for every positive integer $k$. Let $\mathcal{W}(c_k)$ be this set. The first goal of this article is to determine the topological structure of $\mathcal{W}(c_k)$. Under some conditions on $(c_k)_{k\in \mathbb{N}}$, we reveal that $\mathcal{W}(c_k)\cap [0,a]$ is homeomorphic to the Cantor middle third set for some $a$. The second goal is to propose an algebraically independent subset of $\mathcal{W}(c_k)$ if $c_k$ is rapidly increasing. As a corollary, we disclose that the minimum of $\mathcal{W}(k)$ is transcendental. In addition, we apply the main result to the set of $A>1$ such that the integer part of $A^{3^{k!}}$ is always a prime number. As a consequence, we give a certain infinite subset of this set which is algebraically independent. Furthermore, we also get results on the rational approximation, $\mathbb{Q}$-linear independence, and numerical calculations of elements in $\mathcal{W}(c_k)$.

Moduli stabilization in asymptotic flux compactifications

We present a novel strategy to systematically study complex-structure moduli stabilization in Type IIB and F-theory flux compactifications. In particular, we determine vacua in any asymptotic regime of the complex-structure moduli space by exploiting powerful tools of asymptotic Hodge theory. In a leading approximation the moduli dependence of the vacuum conditions are shown to be polynomial with a dependence given by sl(2)-weights of the fluxes. This simple algebraic dependence can be extracted in any asymptotic regime, even though in nearly all asymptotic regimes essential exponential corrections have to be present for consistency. We give a pedagogical introduction to the sl(2)-approximation as well as a detailed step-by-step procedure for constructing the corresponding Hodge star operator. To exemplify the construction, we present a detailed analysis of several Calabi-Yau three- and fourfold examples. For these examples we illustrate that the vacua in the sl(2)-approximation match the vacua obtained with all polynomial and essential exponential corrections rather well, and we determine the behaviour of the tadpole contribution of the fluxes. Finally, we discuss the structure of vacuum loci and their relations to several swampland conjectures. In particular, we comment on the realization of the so-called linear scenario in view of the tadpole conjecture.

Algebraic independence of the Carlitz period and its hyperderivatives

This paper deals with the fundamental period π˜ of the Carlitz module. The main theorem states that the Carlitz period and all its hyperderivatives are algebraically independent over the base field Fq(θ). Our approach also reveals a connection of these hyperderivatives with the coordinates of a period lattice generator of the tensor powers of the Carlitz module which was already observed by M. Papanikolas in a yet unpublished paper. Namely, these coordinates can be obtained by explicit polynomial expressions in π˜ and its hyperderivatives. Papanikolas also gave various presentations of these expressions which we also prove here.

On the algebraic dependence of holonomic functions

We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular, if f and g satisfy linear differential equations and are algebraically dependent, we give conditions on the differential Galois group associated to f guaranteeing that g is a polynomial in f. We apply this to hypergeometric functions and iterated integrals.

Algebraic dependence and finiteness problems of differentiably nondegenerate meromorphic mappings on Kähler manifolds

Abstract Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball 𝔹 m (R 0) in ℂ m (0 &lt; R 0 +∞). Our first aim in this paper is to study the algebraic dependence problem of differentiably meromorphic mappings. We will show that if k differentibility nonde-generate meromorphic mappings f 1, …, fk of M into ℙ n (ℂ) (n ≥ 2) satisfying the condition (Cρ ) and sharing few hyperplanes in subgeneral position regardless of multiplicity then f 1 Λ … Λ fk 0. For the second aim, we will show that there are at most two different differentiably nondegenerate meromorphic mappings of M into ℙ n (ℂ) sharing q (q ∼ 2N − n + 3 + O(ρ)) hyperplanes in N−subgeneral position regardless of multiplicity. Our results generalize previous finiteness and uniqueness theorems for differentiably meromorphic mappings of ℂ m into ℙ n (ℂ) and extend some previous results for the case of mappings on Kähler manifold.

Specific Heat of the Kagome Antiferromagnet Herbertsmithite in High Magnetic Fields

Measuring the specific heat of herbertsmithite single crystals in high magnetic fields (up to $34$ T) allows us to isolate the low-temperature kagome contribution while shifting away extrinsic Schottky-like contributions. The kagome contribution follows an original power law $C_{p}(T\rightarrow0)\propto T^{\alpha}$ with $\alpha\sim1.5$ and is found field-independent between $28$ and $34$ T for temperatures $1\leq T\leq4$ K. These are serious constrains when it comes to replication using low-temperature extrapolations of high-temperature series expansions. We manage to reproduce the experimental observations if about $10$ % of the kagome sites do not contribute. Between $0$ and $34$ T, the computed specific heat has a minute field dependence then supporting an algebraic temperature dependence in zero field, typical of a critical spin liquid ground state. The need for an effective dilution of the kagome planes is discussed and is likely linked to the presence of copper ions on the interplane zinc sites. At very low temperatures and moderate fields, we also report some small field-induced anomalies in the total specific heat and start to elaborate a phase diagram.

Semi-abelian analogues of Schanuel conjecture and applications

In this article we study semi-abelian analogues of Schanuel conjecture. As showed by the first author, Schanuel conjecture is equivalent to the Generalized Period conjecture applied to 1-motives without abelian part. Extending her methods, the second, the third and the fourth authors have introduced the abelian analogue of Schanuel conjecture as the Generalized Period conjecture applied to 1-motives without toric part. As a first result of this paper, we define the semi-abelian analogue of Schanuel conjecture as the Generalized Period conjecture applied to 1-motives.C. Cheng et al. proved that Schanuel conjecture implies the algebraic independence of the values of the iterated exponential and the values of the iterated logarithm, answering a question of M. Waldschmidt. The second, the third and the fourth authors have investigated a similar question in the setup of abelian varieties: the Weak Abelian Schanuel conjecture implies the algebraic independence of the values of the iterated abelian exponential and the values of an iterated generalized abelian logarithm. The main result of this paper is that a Relative Semi-abelian conjecture implies the algebraic independence of the values of the iterated semi-abelian exponential and the values of an iterated generalized semi-abelian logarithm.

Autonomous first order differential equations

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a ‘complete’ answer, obtained independently of model theoretic results on differentially closed fields. Instead, the geometry of curves and generalized Jacobians provides the key ingredient. Classification and formal solutions of autonomous equations are treated. The results are applied to answer a question on D n D^n -finiteness of solutions of first order differential equations.

Separation for isometric group actions and hyperimaginary independence

We generalize P. M. Neumann’s Lemma to the setting of isometric actions on metric spaces and use it to prove several results in continuous model theory related to algebraic independence. In particular, we show that algebraic independence satisfies the ful

Algebraic independence of certain entire functions of two variables generated by linear recurrences

We construct an entire function of two variables having the property that its values and its partial derivatives of any order at any distinct algebraic points are algebraically independent. Such an entire function is obtained as a corollary to the main th

Second main theorem for meromorphic mappings intersecting moving targets on parabolic manifolds

In this paper, we establish a new second main theorem for meromorphic mappings from $M$ into $\mathbb{P}(V)$ intersecting moving targets $g_{j}:M\rightarrow\mathbb{P}(V^{\ast}),\ 1\leq j\leq q,$ where $M$ is a parabolic manifold and $V$ is a Hermitian vector space. As an application, we prove the algebraic dependence problem for meromorphic mappings with moving targets in general position.