Transcendental Number Theory Research Articles

Algebraic independence of elements from $ \mathbb{C}_p $ over $ \mathbb{Q}_p $ , I

Until now there are only very few contributions to the problem in the title, probably because it has no analogue in classical transcendence theory, \( \mathbb{C} \) being a quadratic extension of \( \mathbb{R} \). The main aim of this article is to prove a new criterion giving sufficient conditions for the algebraic independence of finitely many elements from \( \mathbb{C}_p \) over \( \mathbb{Q}_p \), if these elements are defined by certain p-adic series. Some applications of the criterion are also presented.

Dynamics on K3 surfaces: Salem numbers and Siegel disks

This paper presents the first examples of K3 surface automorphisms \(f : X \rightarrow X\) with Siegel disks (domains on which f acts by an irrational rotation). The set of such examples is countable, and the surface \(X\) must be non-projective to carry a Siegel disk. These automorphisms are synthesized from Salem numbers of degree 22 and trace −1, which play the role of the leading eigenvalue for \(f*|H^2(X)\). The construction uses the Torelli theorem, the Atiyah-Bott fixed-point theorem and results from transcendence theory.

Metaphorical Language: Seeing and Hearing with the Heart

Metaphorical language is the communication tool of poetry therapy. Now a metatheory, “The Theory of Transcendence” supports the use of metaphor as the method of making invisible realities “visible.” The poetry therapist will benefit from this theoretical framework that encourages communications across disciplines and encourages the gnosis (knowing) of why a specific metaphor is chosen.

Entropy and the Canonical Height

The height of an algebraic number in the sense of Diophantine geometry is a measure of arithmetic complexity. There is a well-known relationship between the entropy of automorphisms of solenoids and classical heights. We consider an elliptic analogue of this relationship, which involves two novel features. First we consider the introduction of a notion of entropy for sequences of transformations. Second, we consider the recognition of canonical local heights as integrals over the closure of the torsion subgroup of the curve (an elliptic Jensen formula). A sequence of transformations is defined for which there is a canonical arithmetically defined quotient whose entropy is the canonical height and in which the fibre entropy is accounted for by canonical local heights at primes of singular reduction, yielding a dynamical interpretation of singular reduction. This system is related to local systems whose entropy coincides with the canonical local height up to sign. The proofs use transcendence theory, a strong form of Siegel's theorem, and an elliptic analogue of Jensen's formula.

Quantitative Results on Algebraic Independence of Some Numbers with Good Approximations

In this paper, we shall establish transcendence measures of several numbers with good approximations by algebraic numbers. This generalizes the results of Caveny (Proc. Amer. Math. Soc.120 (1994) 349–357) and Tubbs (Proc. Amer. Math. Soc.109 (1990) 297–304), and gives a quantitative version of Ably's result (C.R. Acad. Sci. Paris1(313), serie I (1991) 653–655).

A measure of simultaneous approximation for quasi-modular functions

We improve on a recent result of Saradha giving a transcendence measure for the quotient ŋω of a period of an elliptic curve defined over ℚ by its associated quasi-period. In an (“almost” successful) attempt to include in a single measure both this result and that obtained by Reyssat in 1980, we blend into the “modular method” ideas related to modular but also hypergeometric functions, as appearing e.g. in Andre's work, as well as some Galois considerations.

Model-checking continuous-time Markov chains

We present a logical formalism for expressing properties of continuous-time Markov chains. The semantics for such properties arise as a natural extension of previous work on discrete-time Markov chains to continuous time. The major result is that the verification problem is decidable; this is shown using results in algebraic and transcendental number theory.

Algebraic independence of the values ofE-functions at singular points and Siegel’s conjecture

We prove a generalization of Shidlovskii’s theorem on the algebraic independence of the values ofE-functions satisfying a system of linear differential equations that is well known in the theory of transcendental numbers. We consider the case in which the values ofE-functions are taken at singular points of these systems. Using the obtained results, we prove Siegel’s conjecture that, for the case of first-order differential equations, anyE-function satisfying a linear differential equation is representable as a polynomial in hypergeometricE-functions.

Transcendence measure for η/ω

Transcendence measure for η/ω

On the measure of approximation of the number π by algebraic numbers

By means of the method of the Laurent interpolation determinant, it is proved that, if ζ is an algebraic number, the real numbersd andL satisfy the inequalitiesd≥degζ,L≥L(ζ), andL≥3, and the numberd is sufficiently large, then the inequality $$|\pi - \varsigma | \geqslant \exp ( - 21.4708d \cdot (\log L + d \cdot \log d) \cdot (1 + \log d))$$ holds. The constant 21.4708 in the above estimate for the measure of transcendence of the number π is the best among the known values.

Irrationality Results for Values of Generalized Tschakaloff Series

Arithmetical properties of values of the entire functionTq(x)=∑∞n=0xn/q(1/2)n(n+1), whereqis a parameter, |q|>1, were first studied by L. Tschakaloff (1921,Math. Ann.80, 62–74;84, 100–114). In this paper we introduce a generalization ofTq(x), given by (1.3), and prove the irrationality results for the values of (1.3) at rational points (see Theorem and Corollaries at the end of Section 1). One of the essential tools in the proof is a variant of Mahler's transcendence method, due to J. H. Loxton and A. J. van der Poorten (1977,in“Transcendence Theory: Advances and Applications,” pp. 211–226, Academic Press, San Diego).

What is a Closed-Form Number?

If a student asks for an antiderivative of exp(x^2), there is a standard reply: the answer is not an elementary function. But if a student asks for a closed-form expression for the real root of x = cos(x), there is no standard reply. We propose a definition of a closed-form expression for a number (as opposed to a *function*) that we hope will become standard. With our definition, the question of whether the root of x = cos(x) has a closed form is, perhaps surprisingly, still open. We show that Schanuel's conjecture in transcendental number theory resolves questions like this, and we also sketch some connections with Tarski's problem of the decidability of the first-order theory of the reals with exponentiation. Many (hopefully accessible) open problems are described.

Determining the small solutions to 𝑆-unit equations

In this paper we generalize the method of Wildanger for finding small solutions to unit equations to the case of S S -unit equations. The method uses a minor generalization of the LLL based techniques used to reduce the bounds derived from transcendence theory, followed by an enumeration strategy based on the Fincke-Pohst algorithm. The method used reduces the computing time needed from MIPS years down to minutes.

Quantum Computability

In this paper some theoretical and (potentially) practical aspects of quantum computing are considered. Using the tools of transcendental number theory it is demonstrated that quantum Turing machines (QTM) with rational amplitudes are sufficient to define the class of bounded error quantum polynomial time (BQP) introduced by Bernstein and Vazirani [Proc. 25th ACM Symposium on Theory of Computation, 1993, pp. 11--20, SIAM J. Comput., 26 (1997), pp. 1277--1339]. On the other hand, if quantum Turing machines are allowed unrestricted amplitudes (i.e., arbitrary complex amplitudes), then the corresponding BQP class has uncountable cardinality and contains sets of all Turing degrees. In contrast, allowing unrestricted amplitudes does not increase the power of computation for error-free quantum polynomial time (EQP). Moreover, with unrestricted amplitudes, BQP is not equal to EQP. The relationship between quantum complexity classes and classical complexity classes is also investigated. It is shown that when quantum Turing machines are restricted to have transition amplitudes which are algebraic numbers, BQP, EQP, and nondeterministic quantum polynomial time (NQP) are all contained in PP, hence in ${\rm P}^{#{\rm P}}$ and PSPACE. A potentially practical issue of designing "machine independent" quantum programs is also addressed. A single ("almost universal") quantum algorithm based on Shor's method for factoring integers is developed which would run correctly on almost all quantum computers, even if the underlying unitary transformations are unknown to the programmer and the device builder.

On the measure of algebraic independence of values of derivatives of ap-adic modular function

The measure of the relative transcendence for the values of the derivatives of ap-adic Weyl-Lutz modular functionJ(q) at algebraic points is estimated.

A transcendence measure for π2

A new proof of the fact that π2 is transcendental is proposed. A modification of Hermite's method for an expressly constructed Nikishin system is used. The Beukers integral, which was previously used to prove Apéry's theorem on the irrationality of ζ(2) and ζ(3) is a special case of this construction.

On the algebraic structure of functional matrices of special form

Algebraic properties of functional matrices arising in the constuction of graded Pade approximations are established. This construction plays an important role in the theory of transcendental numbers.

On the Partial Generalization of the Measure of Transcendence of Some Formal Laurent Series

In this work, we determine the trallscendence measure of the formal Laurent series "that" [...]

A Metrical Result on Transcendence Measures in Certain Fields

We give transcendence measures for almost all elements of Fq((1/T)), the field of formal power series in 1/Tover a finite field Fqwithqelements.

A transcendence measure for $\pi^2$

A transcendence measure for $\pi^2$