Differential Algebraic Varieties Research Articles

Bertini theorems for differential algebraic geometry

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini’s theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties.

Finiteness theorems on hypersurfaces in partial differential-algebraic geometry

Hrushovski's generalization and application of Jouanolou (1978) [9] is here refined and extended to the partial differential setting with possibly nonconstant coefficient fields. In particular, it is shown that if X is a differential-algebraic variety over a partial differential field F that is finitely generated over its constant field F0, then there exists a dominant differential-rational map from X to the constant points of an algebraic variety V over F0, such that all but finitely many codimension one subvarieties of X over F arise as pull-backs of algebraic subvarieties of V over F0. As an application, it is shown that the algebraic solutions to a first order algebraic differential equation over C(t) are of bounded height, answering a question of Eremenko. Two expected model-theoretic applications to DCF0,m are also given: 1) Lascar rank and Morley rank agree in dimension two, and 2) dimension one strongly minimal sets orthogonal to the constants are ℵ0-categorical. A detailed exposition of Hrushovski's original (unpublished) theorem is included, influenced by Ghys (2000) [5].

An upper bound for the order of a differential algebraic variety

we present an upper bound for the order of each irreducible component of finite order of a differential algebraic variety V in the context of partial differential fields of characteristic zero. This bound is constructed recursively in terms of explicit data obtained from V .

A Differential Analog of the Noether Normalization Lemma

In this paper, we prove the following differential analog of the Noether normalization lemma: for every $d$-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map onto the $d$-dimensional affine space. Equivalently, for every integral differential algebra $A$ over differential field of zero characteristic there exist differentially independent $b_1, \ldots, b_d$ such that $A$ is differentially algebraic over subalgebra $B$ differentially generated by $b_1, \ldots, b_d$, and whenever $\mathfrak{p} \subset B$ is a prime differential ideal, there exists a prime differential ideal $\mathfrak{q} \subset A$ such that $\mathfrak{p} = B \cap \mathfrak{q}$. We also prove the analogous theorem for differential algebraic varieties over the ring of formal power series over an algebraically closed differential field and present some applications to differential equations.

On Linear Dependence Over Complete Differential Algebraic Varieties

We extend Kolchin's results from [12] on linear dependence over the constant points of projective algebraic varieties to linear dependence over arbitrary complete differential algebraic varieties. We show that in this more general setting, the notion of linear dependence still has necessary and sufficient conditions given by the vanishing of a certain system of differential-polynomials equations. We also discuss some conjectural questions around completeness and the catenary problem.

Bounding the size of a finite differential algebraic variety

Bounding the size of a finite differential algebraic variety

Indecomposability for differential algebraic groups

We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for differential algebraic groups. The theorem establishes a sufficient criterion for the subgroup of a differential algebraic group generated by an infinite family of subvarieties to be a differential algebraic subgroup. This theorem is used for various definability results. For instance, we show that every noncommutative almost simple differential algebraic group is perfect, solving a problem of Cassidy and Singer. We also establish numerous bounds on Kolchin polynomials, some of which seem to be of a nature not previously considered in differential algebraic geometry; in particular, we establish bounds on the Kolchin polynomial of the generators of the differential field of definition of a differential algebraic variety.

Completeness in partial differential algebraic geometry

This paper is part of the model theory of fields of characteristic 0, equipped with m commuting derivation operators (DCF0,m). It continues to partial differential fields work begun by Wai-Yan Pong, who treated the case m=1. We study the concept of completeness in differential algebraic geometry, applying methods of model theory and differential algebra. Our central tool in applying the valuative criterion developed in differential algebra by E.R. Kolchin, Peter Bloom, and Sally Morrison is a fundamental theorem in classical elimination theory due to the model theorist Lou van den Dries. We use this valuative criterion to give a new family of complete differential algebraic varieties. In addition to completeness, we prove some embedding theorems for differential algebraic varieties of arbitrary differential transcendence degree. As a special case, we show that every differential algebraic subvariety of the projective line which has Lascar rank less than ωm can be embedded in the affine line.

New critical points for the liquid phase and the construction of thermodynamics depending on the interaction potential

A mechanism for the inheritance of properties of spectra by differential spectra is developed and applied to prove geometric properties of morphisms of differential algebraic varieties.

Inheritance of properties of spectra

A mechanism for the inheritance of properties of spectra by differential spectra is developed and applied to prove geometric properties of morphisms of differential algebraic varieties.

Jet spaces of varieties over differential and difference fields

We prove some new structural results on finite-dimensional differential algebraic varieties and difference algebraic varieties in characteristic zero, using elementary methods involving jet spaces. Some partial results and problems are given in the positive characteristic cases. The impact of these methods and results on proofs of the Mordell-Lang conjecture for function fields will also be discussed.

Well-Ordering of Certain Numerical Polynomials

An ordering is introduced in the set of numerical polynomials (in one variable) which is then shown to induce a well-ordering on a certain subset of numerical polynomials, namely those which occur as the differential dimension polynomials of differential algebraic varieties, or equivalently, those which come from initial subsets of ${{\text {N}}^m}$.

Well-ordering of certain numerical polynomials

An ordering is introduced in the set of numerical polynomials (in one variable) which is then shown to induce a well-ordering on a certain subset of numerical polynomials, namely those which occur as the differential dimension polynomials of differential algebraic varieties, or equivalently, those which come from initial subsets of ${{\text {N}}^m}$.