Non-zero Algebraic Numbers Research Articles

Algebraic independence of values of quasi-modular forms

In this paper, we discuss the algebraic independence of special values of quasi-modular forms. We show that for three algebraically independent quasi-modular forms with algebraic Fourier coefficients, if one of e2πiz and their values at z in the upper half plane is a non-zero algebraic number, then the other three numbers are algebraically independent over Q. We also provide a condition for determining when quasi-modular forms are algebraically independent. Moreover, based on the theorem proved in the paper, we prove the algebraic independence of values of derivatives of modular forms with algebraic Fourier coefficients.

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.

Undecidable arithmetic properties of solutions of Fredholm integral equations

A basic problem in transcendental number theory is to determine the arithmetic properties of values of special functions. Many special functions, such as Bessel functions and certain hypergeometric functions, are E-functions which are a natural generalization of the exponential function and satisfy certain linear differential equations. In this case, there exists an algorithm which determines if f(α) is transcendental or algebraic if f(z) is an E-function and α∈Q‾⁎ is a non-zero algebraic number. In this paper, we consider the analogous question when f(z) satisfies an integral equation, in particular, a Fredholm integral equation of the first or second kind where the kernel and forcing term satisfy strong arithmetic properties. We show that in both periodic and non-periodic cases, there exists no algorithm to determine if f(0)∈Q is rational. Our results are an application of the undecidability of the Generalized Collatz Problem due to Conway [6].

Entire functions with undecidable arithmetic properties

A basic problem in transcendental number theory is to determine the arithmetic properties of analytic functions of the form f(z)=∑k=0∞akzk where the coefficients ak∈K belong to an algebraic number field. In particular, one of the most basic problems is to determine if f(α) is algebraic or transcendental for non-zero algebraic arguments α. For example, if f(z) is a transcendental Mahler function, then under generic conditions f(α) is transcendental for all non-zero algebraic numbers with |α|<1. Also, if f(z) is an E-function, then there exist algorithms which completely determine the arithmetic properties of f(n)(α) for non-zero algebraic numbers α. In contrast to these and other algorithmic results, we construct three functions f(z), g(z), and h(z) with computable rational coefficients for which no algorithms exist that determine if f(n)∈Q, g(n)(1)∈Q, or ∫01h(z)zndz∈Q for integral n≥0. Our results are an application of an undecidable variant of the Collatz Problem due to Lehtonen [9].

A note on trace of powers of algebraic numbers

We study pairs of non-zero algebraic numbers (α,λ) such that TrQ(α,λ)/Q(λαn)∈Z for n lying in suitable subsets I of N. In this short note, we give a satisfactory account to these circle of questions. This theme arises naturally in the study of distribution of exponential sequences modulo one. Finally we indicate that the analogous results do not hold for global fields in positive characteristic.

Approximation by Shifts of Compositions of Dirichlet L-Functions with the Gram Function

In this paper, a joint approximation of analytic functions by shifts of Dirichlet L-functions L ( s + i a 1 t τ , χ 1 ) , … , L ( s + i a r t τ , χ r ) , where a 1 , … , a r are non-zero real algebraic numbers linearly independent over the field Q and t τ is the Gram function, is considered. It is proved that the set of their shifts has a positive lower density.

Generalized GCD for toric Fano varieties

We study the greatest common divisor problem for torus invariant blowing-up morphisms of nonsingular toric Fano varieties. Our main result applies the theory of Okounkov bodies together with an arithmetic form of Cartan's Second Main theorem, which has been established by Ru and Vojta. It also builds on Silverman's geometric concept of greatest common divisor. As a special case of our results, we deduce a bound for the generalized greatest common divisor of pairs of nonzero algebraic numbers.

On Mahler s Sinumbers, Tinumbers, and Uinumbers

In this work, we consider some power series with algebraic coefficients from a certain algebraic number field, whose radiuses of convergence are infinite. We show that under certain conditions these series take transcendental values at non-zero algebraic number arguments, and we determine the classes to which these transcendental values belong in Mahler’s classification. Then we consider these series for certain Liouville number arguments and obtain similar results.

Algebraic approximations to linear combinations of powers: An extension of results by Mahler and Corvaja–Zannier

For every complex number $x$, let $\Vert x\Vert _{\mathbb {Z}}:=\min \{|x-m|:\ m\in \mathbb {Z}\}$. Let $K$ be a number field, let $k\in \mathbb {N}$, and let $\alpha _1,\ldots ,\alpha _k$ be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of $\theta \in (0,1)$ such that there are infinitely many tuples $(n,q_1,\ldots ,q_k)$ satisfying $\Vert q_1\alpha _1^n+\cdots +q_k\alpha _k^n\Vert _{\mathbb {Z}}<\theta ^n$ where $n\in \mathbb {N}$ and $q_1,\ldots ,q_k\in K^*$ have small logarithmic height compared to $n$. In the special case when $q_1,\ldots ,q_k$ have the form $q_i=qc_i$ for fixed $c_1,\ldots ,c_k$, our work yields results on algebraic approximations of $c_1\alpha _1^n+\cdots +c_k\alpha _k^n$ of the form $\frac {m}{q}$ with $m\in \mathbb {Z}$ and $q\in K^*$ (where $q$ has small logarithmic height compared to $n$). Various results on linear recurrence sequences also follow as an immediate consequence. The case where $k=1$ and $q_1$ is essentially a rational integer was obtained by Corvaja and Zannier and settled a long-standing question of Mahler. The use of the Subspace Theorem based on work of Corvaja–Zannier, together with several modifications, plays an important role in the proof of our results.

A Weil Banach Algebra for Multiplicative Algebraic Numbers

We present a generalization of the construction of Allcock and Vaaler in [AV09] which established that the Weil height provides non-zero algebraic numbers modulo torsion with the structure of a normed vector space and construct its completion as L1 of a locally compact measure space. In the present work we show that this space is naturally a Banach algebra and describe its structure and a method for computing the Banach multiplication explicitly using the theory of harmonic analysis on compact groups.

Algebraic independence of the values of the Hecke–Mahler series and its derivatives at algebraic numbers

We show that the Hecke–Mahler series, the generating function of the sequence [Formula: see text] for [Formula: see text] real, has the following property: Its values and its derivatives of any order, at any nonzero distinct algebraic numbers inside the unit circle, are algebraically independent if [Formula: see text] is a quadratic irrational number satisfying a suitable condition.

Algebraic independence of the values of functions satisfying Mahler type functional equations under the transformation represented by a power relatively prime to the characteristic of the base field

We give positive characteristic analogues of complex entire functions having remarkable property that their values as well as their derivatives of any order at any nonzero algebraic numbers are algebraically independent. These results are obtained by establishing a criterion for the algebraic independence of the values of Mahler functions as well as that of the algebraic independence of the Mahler functions themselves over any function fields of positive characteristic.

Counting exceptional points for rational numbers associated to the Fibonacci sequence

If $\alpha$ is a non-zero algebraic number, we let $m(\alpha)$ denote the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the $t$-metric Mahler measure, denoted $m_t(\alpha)$. For fixed $\alpha\in \bar{\mathbb Q}$, the map $t\mapsto m_t(\alpha)$ is continuous, and moreover, is infinitely differentiable at all but finitely many points, called {\it exceptional points} for $\alpha$. It remains open to determine whether there is a sequence of elements $\alpha_n\in \bar{\mathbb Q}$ such that the number of exceptional points for $\alpha_n$ tends to $\infty$ as $n\to \infty$. We utilize a connection with the Fibonacci sequence to formulate a conjecture on the $t$-metric Mahler measures. If the conjecture is true, we prove that it is best possible and that it implies the the existence of rational numbers with as many exceptional points as we like. Finally, with some computational assistance, we resolve various special cases of the conjecture that constitute improvements to earlier results.

Schanuel's theorem for heights defined via extension fields

Let k be a number field, let ✓ be a nonzero algebraic number, and let H(·) be the Weil height on the algebraic numbers. In response to a question by T. Loher and D. W. Masser, we prove an asymptotic formula for the number of ↵ 2 k with H(↵✓)  X, and we analyze the leading constant in our asymptotic formula. In particular, we prove a sharp upper bound in terms of the classical Schanuel constant. We also prove an asymptotic counting result for a new class of height functions defined via extension fields of k with a fairly explicit error term. This provides a conceptual framework for Loher and Masser’s problem and generalizations thereof. Finally, we establish asymptotic counting results for varying ✓, namely, for the number of pp↵ of bounded height, where ↵ 2 k and p is any rational prime inert in k.

Algebraic Independence of Mahler Functions via Radial Asymptotics

We present a new method for algebraic independence results in the context of Mahler's method. In particular, our method uses the asymptotic behavior of a Mahler function |$f(z)$| as |$z$| goes radially to a root of unity to deduce algebraic independence results about the values of |$f(z)$| at algebraic numbers. We apply our method to the canonical example of a degree two Mahler function; that is, we apply it to |$F(z)$|⁠, the power series solution to the functional equation |$F(z)-(1+z+z^2)F(z^4)+z^4F(z^{16})=0$|⁠. Specifically, we prove that the functions |$F(z)$|⁠, |$F(z^4)$|⁠, |$F'(z)$|⁠, and |$F'(z^4)$| are algebraically independent over |$\mathbb {C}(z)$|⁠. An application of a celebrated result of Ku. Nishioka then allows one to replace |$\mathbb {C}(z)$| by |$\mathbb {Q}$| when evaluating these functions at a nonzero algebraic number |$\alpha $| in the unit disc.Communicated by Prof. Umberto Zannier

Metric heights on an Abelian group

Suppose $m(\alpha)$ denotes the Mahler measure of the non-zero algebraic number $\alpha$. For each positive real number $t$, the author studied a version $m_t(\alpha)$ of the Mahler measure that has the triangle inequality. The construction of $m_t$ is generic and may be applied to a broader class of functions defined on any Abelian group $G$. We prove analogs of known results with an abstract function on $G$ in place of the Mahler measure. In the process, we resolve an earlier open problem stated by the author regarding $m_t(\alpha)$.

Algebraic Numbers Satisfying Polynomials with Positive Rational Coefficients

A theorem of Dubickas, affirming a conjecture of Kuba, states that a nonzero algebraic number is a root of a polynomial f with positive rational coefficients if and only if none of its conjugates is a positive real number. A certain quantitative version of this result, yielding a growth factor for the coefficients of f similar to the condition of the classical Eneström-Kakeya theorem of such polynomial, is derived. The bound for the growth factor so obtained is shown to be sharp for some particular classes of algebraic numbers.

On the values of $G$-functions

In this paper we study the set $\mathbf{G}$ of values at algebraic points of analytic continuations of $G$-functions (in the sense of Siegel). This subring of $\mathbb{C}$ contains values of elliptic integrals, multiple zeta values, and values at algebraic points of generalized hypergeometric functions ${p+1}F\_p$ with rational coefficients. Its group of units contains non-zero algebraic numbers, $\pi$, $\Gamma(a/b)^b$ and $B(x,y)$ (with $a,b\in \mathbb{Z}$ such that $a/b\not\in \mathbb Z$, and $x,y\in\mathbb{Q}$ such that $B(x,y)$ exists and is non-zero). We prove that for any $\xi \in \mathbf{G}$, both $\operatorname{Re} \xi$ and $\mathrm{Im} , \xi$ can be written as $f(1)$, where $f$ is a $G$-function with rational coefficients of which the radius of convergence can be made arbitrarily large. As an application, we prove that quotients of elements of $\mathbf{G} \cap \mathbb{R}$ are exactly the numbers which can be written as limits of sequences $a\_n/b\_n$, where $\sum{n=0}^{\infty} a\_n z^n$ and $\sum\_{n=0}^{\infty} b\_n z^n$ are $G$-functions with rational coefficients. This result provides a general setting for irrationality proofs in the style of Apéry for $\zeta(3)$, and gives answers to questions asked by T. Rivoal in “Approximations rationnelles des valeurs de la fonction Gamma aux rationnels: le cas des puissances”, Acta Arith. 142 (2010), no. 4, 347–365.

On the algebraic independence of certain Mahlerʼs U-numbers

Abstract We examine the algebraic independence over Q of certain Mahlerʼs U-numbers appearing as the values of certain generalized lacunary power series at non-zero algebraic number arguments.

The Infimum in the Metric Mahler Measure

AbstractDubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number α by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure.