Algebraic independence

This paper is concerned with algebraic independence in structures that are relatively simple for their size. It is shown that for κ a limit cardinal, if a structure of power at least κ is ∞ω-equivalent to a structure of power less than κ, then must contain an infinite set of algebraically independent elements. The same method of proof yields the fact that if σ is an Lω1ω-sentence (not necessarily complete) and σ has a model of power ℵω then some model of σ contains an infinite algebraically independent set.All structures are assumed to be of countable similarity type. Letters , etc. will be used to denote either a structure or the universe of the structure. If X ⊆ , the algebraic closure of X (in ), denoted by Cl(X), is the union of all finite sets that are weakly definable (in ) by Lωω-formulas with parameters from X. A set S is algebraically independent if for each a in S, a ∉ Cl(S – {a}). An algebraically independent set is sometimes called a “free” set (in [3] and [4], for example).It is known (see [5]) that any structure of power ℵn must have a set of n algebraically independent elements, and there are structures of power ℵn with no independent set of size n + 1. In power ℵω every structure will have arbitrarily large finite algebraically independent sets. However, it is consistent with ZFC that some models of power ℵω do not have any infinite algebraically independent set. Devlin [4] showed that if V = L, then for any cardinal κ, if every structure of power κ has an infinite algebraically independent set, then κ has a certain large cardinal property that ℵω can never possess.

В настоящей работе получены примеры алгебраических тождеств между фундаментальнымиматрицами обобщённых гипергеометрических уравнений. В некоторых случаях эти тождествапорождают все алгебраические соотношения между компонентами решенийгипергеометрических уравнений.Обобщённые гипергеометрические функции (см. [1-5]) - это функции вида$${}_l\varphi_{q}(z)={}_l\varphi_{q}(\vec \nu;\vec\lambda;z)={}_{l+1}F_{q}\left(\left.{1,\nu_1,\dots,\nu_l\atop\lambda_1,\dots,\lambda_q}\right|z\right)=\sum_{n=0}^\infty \frac{(\nu_1)_n\dots (\nu_l)_n}{(\lambda_1)_n\dots(\lambda_{q})_n} z^n,$$где $0\leqslant l\leqslant q$, $\; (\nu)_0=1, \; (\nu)_n=\nu(\nu+1)\dots (\nu+n-1)$,$\;\vec\nu=(\nu_1,\dots,\nu_l)\in {\mathbb C}^l$, $\;\vec \lambda\in({\mathbb C}\setminus{\mathbb Z^-})^q$.Функция ${}_l\varphi_{q}(\vec \nu;\vec\lambda;z)$ удовлетворяет(обобщённому) гипергеометрическому дифференциальному уравнению$${L}(\vec \nu;\vec\lambda;z)\;y =(\lambda_1-1)\dots(\lambda_q-1),$$где$${L}(\vec \nu;\vec\lambda;z)\equiv \left( \prod_{j=1}^q(\delta+\lambda_j-1)-z\prod_{k=1}^l(\delta+\nu_k) \right),\label{d1122} \quad \delta=z\frac{d}{dz}.$$В теории трансцендентных чисел одним из основных методов являетсяметод Зигеля-Шидловского (см. [4], [5]), которыйпозволяет доказывать трансцендентность и алгебраическую независимостьзначений целых функций некоторого класса, включающего в себяфункции ${}_l\varphi_{q}(\alpha z^{q-l})$, при условииалгебраической независимости этих функций над ${\mathbb C}(z)$.В статье [6] Ф. Бейкерсом, В. Браунвеллом и Г. Хекманом быливведены важные для установления алгебраической зависимости инезависимости функций понятия коградиентности и контрградиентностидифференциальных уравнений (фактически эти понятия возникли ранеев статье Е. Колчина [7]).Настоящая работа посвящена подробному доказательству и дальнейшемуразвитию результатов о коградиентности и контрградиентности,опубликованных в заметках [8] и [9]. В частности, уточняютсянекоторые результаты статьи [6].

The Siegel-Shidlovskii method is one of the main methods in the theory of transcendental numbers. By using this method, it is possible to establish the transcendency and algebraic independence of the values of entire functions belonging to a certain class (so-called E-functions) provided that these functions are algebraically independent over C(z). There are many examples of E-functions used in mathematics: exp(z), sinz, cosz, sinhz, coshz, Bessel functions, Kummer functions, incomplete gamma-function, generalized hypergeometric functions, and some other special functions. By using the Siegel-Shidlovskii method, it is possible to obtain the lower estimates for the moduli of polynomials from the values of E-functions. These estimates are called algebraic independence measures of the numbers. They serve as the quantitative characteristics of algebraic independence. The problem of obtaining the estimates of algebraic independence measures has long been dealt with by many researchers. The first estimates of measures for the values of exp(z) were obtained by E. Borel and K. Mahler, and for the values of Bessel functions such estimates were obtained by C. Siegel. Estimates of algebraic independence measures for the values of general E-functions were established by S. Lang, A.I. Galochkin, A.B. Shidlovskii, and Yu.V. Nesterenko. The constant appearing in the estimate of a measure is called effective if it can be expressed in terms of the parameters characterizing the considered functions and points at which their values are taken. An estimate of a measure is considered to be effective if it contains only effective constants. Generally speaking, the algebraic independence condition of functions is not sufficient for obtaining effective estimates of algebraic independence measures. Some additional conditions must be satisfied for this. The case in which the main totality of functions is algebraically dependent over C(z) is the most complicated one. In a number of the author’s works, the estimates of algebraic independence measures of the values of E-functions were improved. After 1995, no papers have been published on this topic, although it still remains of issue.

The article takes a look at transcendence and algebraic independence problems, introduces statements and proofs of theorems for some kinds of elements from direct product of 𝑝-adic fields and polynomial estimation theorem. Let Q𝑝 be the 𝑝-adic completion of Q, Ω𝑝 be the completion of the algebraic closure of Q𝑝, 𝑔 = 𝑝1𝑝2 . . . 𝑝𝑛 be a composition of separate prime numbers, Q𝑔 be the 𝑔-adic completion of Q, in other words Q𝑝1 ⊕. . .⊕Q𝑝𝑛. The ring Ω𝑔 ∼=Ω𝑝1⊕...⊕Ω𝑝𝑛, a subring Q𝑔, transcendence and algebraic independence over Q𝑔 are under consideration. Also, hypergeometric series $$𝑓(𝑧) =∞Σ𝑗=0((𝛾1)𝑗 . . . (𝛾𝑟)𝑗)/((𝛽1)𝑗 . . . (𝛽𝑠)𝑗)(𝑧𝑡)^𝑡𝑗 $$, and their formal derivatives are under consideration. Sufficient conditions are obtained under which the values of the series 𝑓(𝛼) and formal derivatives satisfy global relation of algebraic independence, if 𝛼 =∞Σ𝑗=0 𝑎_𝑗𝑔^(𝑟_𝑗), where 𝑎𝑗 ∈ Z𝑔, and non-negative rationals 𝑟𝑗 increase strictly unbounded.

New results on algebraic independence with Mahlerʼs method

Using a simple application of Fubini's theorem, we examine the connection between statistical independence, linear independence of random vectors, and algebraic independence of univariate r.v.'s, where we call a finite set of r.v.'s algebraically independent if they satisfy a non-trivial polynomial relationship only with zero probability. As a consequence, we simplify the derivation of a result of Eaton and Perlman (1973) on the linear independence of random vectors, and settle a matrix equation question of Okamoto (1973) concerning the rank of sample covariance-type matrices $S = XAX'$, where $X$ is $p \times n$, and $A$ is $n \times n$, for the case $n \geq p \geq r = \operatorname{rank}(A)$. We also derive a measure-theoretic version of the classical fact that the elementary symmetric polynomials in $m$ indeterminates are algebraically independent. This has applications to sample moments, $k$-statistics, and $U$-statistics with polynomial kernels.

In previous papers the authors established a method how to decide on the algebraic independence of a set $\{ y_1,\dots ,y_n \}$ when these numbers are connected with a set $\{ x_1,\dots ,x_n \}$ of algebraic independent parameters by a system $f_i(x_1,\dots ,x_n,y_1,\dots ,y_n) =0$ $(i=1,2,\dots ,n)$ of rational functions. Constructing algebraic independent parameters by Nesterenko's theorem, the authors successfully applied their method to reciprocal sums of Fibonacci numbers and determined all the algebraic relations between three $q$-series belonging to one of the sixteen families of $q$-series introduced by Ramanujan. In this paper we first give a short proof of Nesterenko's theorem on the algebraic independence of $\pi$, $e^{\pi\sqrt{d}}$ and a product of Gamma-values $\Gamma (m/n)$ at rational points $m/n$. Then we apply the method mentioned above to various sets of numbers. Our algebraic independence results include among others the coefficients of the series expansion of the Heuman-Lambda function, the values $P(q^r), Q(q^r)$, and $R(q^r)$ of the Ramanujan functions $P,Q$, and $R$, for $q\in \overline{\ACADQ}$ with $0<|q|<1$ and $r=1,2,3,5,7,10$, and the values given by reciprocal sums of polynomials.

Algebraic independence of values of quasi-modular forms

In this article we show that the coordinates of a period lattice generator of the n -th tensor power of the Carlitz module are algebraically independent, if n is prime to the characteristic. The main part of the paper, however, is devoted to a general construction for t -motives which we call prolongation, and which gives the necessary background for our proof of the algebraic independence. Another incredient is a theorem which shows hypertranscendence for the Anderson-Thakur function \omega(t) , i.e. that \omega(t) and all its hyperderivatives with respect to t are algebraically independent.

Algebraic independence and blackbox identity testing

In the case of systems composed of identical particles, a typical instance in quantum statistical mechanics, the standard approach to separability and entanglement ought to be reformulated and rephrased in terms of correlations between operators from subalgebras localized in spatially disjoint regions. While this algebraic approach is straightforward for bosons, in the case of fermions it is subtler since one has to distinguish between micro-causality, that is the anti-commutativity of the basic creation and annihilation operators, and algebraic independence that is the commutativity of local observables. We argue that a consistent algebraic formulation of separability and entanglement should be compatible with micro-causality rather than with algebraic independence.

We prove that if y ′ ′ = f ( y , y ′ , t , α , β , γ , δ ) is a generic Painlevé equation from among the classes I I I and V I , and if y 1 , … , y n are distinct solutions, then t r · d e g C ( t ) C ( t ) ( y 1 , y 1 ′ , … , y n , y n ′ ) = 2 n , that is y 1 , y 1 ′ , … , y n , y n ′ are algebraically independent over C ( t ) . This improves the results obtained by the author and Pillay and completely proves the algebraic independence conjecture for the generic Painlevé transcendents. In the process, we also prove that any three distinct solutions of a Riccati equation are algebraic independent over C ( t ) , provided that there are no solutions in the algebraic closure of C ( t ) . This uses techniques and results from differential Galois theory and answers a very natural question in the theory of Riccati equations.

In this paper we establish algebraic independence criteria for the values at an algebraic point of Mahler functions each of which satisfies either a multiplicative type of functional equation or an additive one. As application we construct, using a linear recurrence sequence, an entire function defined by an infinite product such that its values as well as its all successive derivatives at algebraic points other than its zeroes are algebraically independent. Zeroes of such an entire function form a subsequence of the linear recurrence sequence. We prove the algebraic independency by reducing those values at algebraic points to those of Mahler functions.

The aim of this paper to draw attention to several aspects of the algebraic dependence in algebras. The article starts with discussions of the algebraic dependence problem in commutative algebras. Then the Burchnall-Chaundy construction for proving algebraic dependence and obtaining the corresponding algebraic curves for commuting differential operators in the Heisenberg algebra is reviewed. Next some old and new results on algebraic dependence of commuting q-difference operators and elements in q-deformed Heisenberg algebras are reviewed. The main ideas and essence of two proofs of this are reviewed and compared. One is the algorithmic dimension growth existence proof. The other is the recent proof extending the Burchnall-Chaundy approach from differential operators and the Heisenberg algebra to the q-deformed Heisenberg algebra, showing that the Burchnall-Chaundy eliminant construction indeed provides annihilating curves for commuting elements in the q-deformed Heisenberg algebras for q not a root of unity. (Less)

Complete C*-independence of operator algebras is introduced. Equivalent characterization is given for C*-subalgebras to be completely independent in terms of maximal tensor product. Besides, the independence of Banach algebras is considered, and we showed that Hahn–Banach independence is a generalization of C*-independence and discussed Hahn–Banach independence in Mn(A), where A is a C*-algebra. Among others, we characterize independence of operator algebras by projective and injective C*-tensor product in terms of simultaneous extensions of completely positive maps.