Omar León Sánchez

Poisson algebras via model theory and differential algebraic geometry

Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier-Moeglin equivalence in finite

On bounds for the effective differential Nullstellensatz

\operatorname{GK}

On parameterized differential Galois extensions

dimension. A weaker version of the Poisson Dixmier-Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.

D-groups and the Dixmier–Moeglin equivalence

Abstract Understanding bounds for the effective differential Nullstellensatz is a central problem in differential algebraic geometry. Recently, several bounds have been obtained using Dicksonian and antichains sequences (with a given growth rate). In the present paper, we make these bounds more explicit and, therefore, more applicable to understanding the computational complexity of the problem, which is essential to designing more efficient algorithms.

On Linear Dependence Over Complete Differential Algebraic Varieties

Abstract We prove some existence results on parameterized strongly normal extensions for logarithmic equations. We generalize a result in Wibmer (2012) [16] . We also consider an extension of the results in Kamensky and Pillay (2014) [4] from the ODE case to the parameterized PDE case. More precisely, we show that if D and Δ are two distinguished sets of derivations and ( K D , Δ ) is existentially closed in ( K , Δ ) , where K is a D ∪ Δ -field of characteristic zero, then every (parameterized) logarithmic equation over K has a parameterized strongly normal extension.

An upper bound for the order of a differential algebraic variety

A differential-algebraic geometric analogue of the Dixmier-Moeglin equivalence is articulated, and proven to hold for

THE MODEL COMPANION OF DIFFERENTIAL FIELDS WITH FREE OPERATORS

D

A New Bound for the Existence of Differential Field Extensions

-groups over the constants. The model theory of differentially closed fields of characteristic zero, in particular the notion of analysability in the constants, plays a central role. As an application it is shown that if

Geometric axioms for differentially closed fields with several commuting derivations

R

Some definable Galois Theory and Examples

is a commutative affine Hopf algebra over a field of characteristic zero, and