The purpose of this expository article is to explain diverse new tools that automata theory provides to tackle transcendence problems in function field arithmetic. We collect and explain various useful results scattered in computer science, formal languages, logic literature and explain how they can be fruitfully used in number theory, dealing with transcendence, refined transcendence and class...

BACKGROUND Although successful aging is most often defined by the absence of disease and disability, older adults consistently report aging successfully even in the presence of chronic illness and functional limitations. A more holistic way of looking at old age suggests transcendence may be an important missing criterion for successful aging. AIM Transcendence, a late life developmental proc...

Although we live in a post-metaphysical age, there is a renewed interest in transcendence, especially at the intersection of philosophy, religion, and theology. There are several reasons for this: among others, the important link that the future (eschatology) has with the unknown or that which lies beyond (transcendence). In this article, this relation between eschatology and transcendence is e...

We develop an algorithm for computing isomorphisms and automorphisms of algebraic function fields of transcendence degree one in characteristic zero and positive characteristic.

OBJECTIVES Gerotranscendence has been conceptualized as a potential development accompanying normal aging. Gerotranscendence is defined as a shift in metaperspective from a materialistic and pragmatic world view to a more cosmic and transcendent one. In the past decade, population-based studies have tested Tornstam's Gerotranscendence Scale. Its Cosmic Transcendence subscale, in particular, eme...

This paper is an extract of a paper that will soon appear in Communications in Algebra. The paper investigates the converse to a theorem which states that a differential domain D finitely generated over a field F of characteristic 0 with no nonzero proper differential ideals will have constants of differentiation C only in the field in question. The converse is known to be false but the questio...

We show that an irreducible component of the Hodge locus of a polarizable variation of Hodge structure of weight 0 on a smooth complex variety X is defined over an algebraically closed subfield k of finite transcendence degree if X is defined over k and the component contains a k-rational point. We also prove a similar assertion for the Hodge locus inside the Hodge bundle if the Hodge bundle to...

Let F be either a number field or a field finitely generated of transcendence degree ≥ 1 over a Hilbertian field of characteristic 0, let F (t) be the rational function field in one variable over F , and let α ∈ Br(F (t)). It is known that there exist infinitely many a ∈ F such that the specialization t→ a induces a specialization α→ α ∈ Br(F ), where α has exponent equal to that of α. Now let ...

We formulate function field analogues for the PinkZilber Conjecture and for the Bounded Height Conjecture. The “special” varieties in our formulation are varieties defined over the constant field. We prove our function field Pink-Zilber Conjecture for all subvarieties, and we prove our function field Bounded Height Conjecture for a certain class of curves. We explain that for our problems the a...

1. Algebraic independence of the two functions ℘(z) and ez. Legendre’s relation η2ω1 − η1ω2 = 2iπ. Proof: integrate ζ(z)dz on a fundamental parallelogram. Application: algebraic independence of the two functions az + bζ(z) and ℘(z). 2. Section § 10.7.2: Morphisms between elliptic curves. The modular invariant. 3. Section § 10.7.3: Endomorphisms of an elliptic curve; complex multiplications. Alg...