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...

Hilbert’s fourteenth problem asks whether the ring of invariants of any representation of a linear algebraic group is finitely generated over the base field. Nagata gave the first counterexample, using a representation of (Ga) 13, where Ga denotes the additive group [22]. In his example, the representation is defined over a field of large transcendence degree over the prime field (of any charac...

Let D be a division ring with an involution and characteristic different from 2. Then, up to a few exceptions, D contains a free pair of symmetric elements provided that (a) it is finite-dimensional and the center has a finite sufficiently large transcendence degree over the prime field, or (b) the center is uncountable, but not algebraically closed in D. Under conditions (a), if the involution...

For a smooth complex projective variety X defined over a number field, we have filtrations on the Chow groups depending of the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we can show that the obtained filtration coincides with the filtration of Green and Griffiths, assuming the Hodge conjecture. In the case the realizations cont...

for analyzing the issue of coincidence of immanence with transcendence in whitehead’s process philosophy, the present paper attempts at the beginning to show that in this process system, especially in the framework of inclusive and comprehensive categories, entities should be interweave and have organic interrelation; because all the actual entities have reaction against each other. so the prer...