Differential Algebraic Geometry 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.

Exploring tropical differential equations

Abstract The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-Archimedean (e.g., p-adic) setting. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right.

On initials and the fundamental theorem of tropical partial differential algebraic geometry

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The fundamental theorem of Tropical Differential Algebraic Geometry and its extensions state that the support of power series solutions of systems of ordinary differential equations (with formal power series coefficients over an uncountable algebraically closed field of characteristic zero) can be obtained either, by solving a so-called tropicalized differential system, or by testing monomial-freeness of the associated initial ideals. Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions, particularly on the existence of solutions.We show here that both of these methods can be generalized to the case of systems of partial differential equations, this is, one can go either with the solution of tropicalized systems, or test monomial-freeness of the ideal generated by the initials when looking for supports of power series solutions of systems of differential equations, regardless the (finite) number of derivatives. The key are the vertex sets of Newton polytopes, upon which relies the definition of both tropical vanishing condition and the initial of a differential polynomial.

Algebraic hyperbolicity for surfaces in smooth projective toric threefolds with Picard rank 2 and 3

Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a very general surface of degree at least five in projective space is algebraically hyperbolic. We are interested in generalizing the study of surfaces in projective space to surfaces in smooth projective toric threefolds with Picard rank 2 or 3. Following Kleinschmidt and Batyrev, we explore the combinatorial description of smooth projective toric threefolds with Picard rank 2 and 3. We then use Haase and Ilten’s method of finding algebraically hyperbolic surfaces in toric threefolds. As a result, we determine many algebraically hyperbolic surfaces in each of these varieties.

Algorithms yield upper bounds in differential algebra

AbstractConsider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the size of the input. We also generalize this to algorithms working with models of good enough theories (including, for example, difference fields).We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

Atiyah’s work on holomorphic vector bundles and gauge theories

The first part of the article surveys Atiyah’s work in algebraic geometry during the 1950s, mainly on holomorphic vector bundles over curves. In the second part we discuss his work from the late 1970s on mathematical aspects of gauge theories, involving differential geometry, algebraic geometry, and topology.

Poisson Dixmier-Moeglin equivalence from a topological point of view

A complex affine Poisson algebra A is said to satisfy the Poisson Dixmier-Moeglin equivalence if the Poisson cores of maximal ideals of A are precisely those Poisson prime ideals that are locally closed in the Poisson prime spectrum P.spec A and if, moreover, these Poisson prime ideals are precisely those whose extended Poisson centers are exactly the complex numbers. In this paper, we provide some topological criteria for the Poisson Dixmier-Moeglin equivalence for A in terms of the poset (P.spec A, ⊆) and the symplectic leaf or core stratification on its maximal spectrum. In particular, we prove that the Zariski topology of the Poisson prime spectrum and of each symplectic leaf or core can detect the Poisson Dixmier-Moeglin equivalence for any complex affine Poisson algebra. Moreover, we generalize the weaker version of the Poisson Dixmier-Moeglin equivalence for a complex affine Poisson algebra proved in [J. Bell, S. Launois, O. L. Sanchez and B. Moosa, Poisson algebras via model theory and differential-algebraic geometry, J. Eur. Math. Soc. (JEMS) 19 (2017), 2019–2049] to the general context of a commutative differential algebra.

Brauer–Clifford group of Azumaya–Poisson (S, H)-algebras

In this paper we extend the notion of the Brauer–Clifford group to the case of an Azumaya–Poisson (S, H)-algebra, when H is a cocommutative Hopf algebra and S is an H-module Poisson algebra. This is the situation that arises in applications with connections to differential geometry, algebraic geometry and quantum groups. These Brauer–Clifford groups turn out to be an example of the Brauer group of a symmetric monoidal category.

Completeness of Finite-Rank Differential Varieties

Differential algebraic geometry offers tantalizing similarities to the algebraic version as well as puzzling anomalies. This thesis builds on results of Kolchin, Blum, Morrison, van den Dries, and Pong to study the problem of completeness for projective differential varieties. The classical fundamental theorem of elimination theory asserts that if V is a projective algebraic variety defined over an algebraically closed field K and W is any algebraic variety defined over K, then the projection VxW -> W takes Zariski-closed sets to Zariski-closed sets. Differential varieties defined by differential polynomial equations over a differentially closed field are more complicated. We give the first example of an incomplete finite-rank differential variety, as well as new instances of complete differential varieties. We also explain how model theory yields multiple versions of Pong's valuative criterion for completeness and reduces the differential completeness problem to one involving algebraic varieties over the complex numbers.

Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems

Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the first case we also discuss a distinguished contact cone structure, arising as the variety of minimal rational tangents. We discuss the repercussion of the aforementioned classification theorems for the geometry of quaternion-K\"ahler manifolds with positive scalar curvature and for the geometry of second-order PDEs imposed on hypersurfaces.

The Hodge conjecture: The complications of understanding the shape of geometric spaces

The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950 and is currently a stimulus for the development of several theories based on geometry, analysis, and mathematical physics. It proposes a natural condition for the existence of complex submanifolds within a complex manifold. Manifolds are the spaces in which geometric objects can be considered. In complex manifolds, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers.

A simplicial foundation for differential and sector forms in tangent categories

Tangent categories provide an axiomatic framework for understanding various tangent bundles and differential operations that occur in differential geometry, algebraic geometry, abstract homotopy theory, and computer science. Previous work has shown that one can formulate and prove a wide variety of definitions and results from differential geometry in an arbitrary tangent category, including generalizations of vector fields and their Lie bracket, vector bundles, and connections. In this paper we investigate differential and sector forms in tangent categories. We show that sector forms in any tangent category have a rich structure: they form a symmetric cosimplicial object. This appears to be a new result in differential geometry, even for smooth manifolds. In the category of smooth manifolds, the resulting complex of sector forms has a subcomplex isomorphic to the de Rham complex of differential forms, which may be identified with alternating sector forms. Further, the symmetric cosimplicial structure on sector forms arises naturally through a new equational presentation of symmetric cosimplicial objects, which we develop herein.

AX–SCHANUEL CONDITION IN ARBITRARY CHARACTERISTIC

We prove a positive characteristic version of Ax’s theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group [Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), 1195–1204]. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax–Schanuel type.

A Mordell–Weil theorem for cubic hypersurfaces of high dimension

Let $X$ be a smooth cubic hypersurface of dimension $n \ge 1$ over the rationals. It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell--Weil theorem for $n=1$, Manin (1968) asked if there exists a finite set $S$ from which all other rational points can be thus obtained. We give an affirmative answer for $n \ge 48$, showing in fact that we can take the generating set $S$ to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on the existence of rational points on the intersection of a quadric and cubic in large dimension, and some elementary ideas from differential geometry, algebraic geometry and numerical analysis.

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 GK 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.

The fundamental theorem of tropical differential algebraic geometry

Let $I$ be an ideal of the ring of Laurent polynomials $K[x_1^{\pm1},\ldots,x_n^{\pm1}]$ with coefficients in a real-valued field $(K,v)$. The fundamental theorem of tropical algebraic geometry states the equality $\text{trop}(V(I))=V(\text{trop}(I))$ between the tropicalization $\text{trop}(V(I))$ of the closed subscheme $V(I)\subset (K^*)^n$ and the tropical variety $V(\text{trop}(I))$ associated to the tropicalization of the ideal $\text{trop}(I)$. In this work we prove an analogous result for a differential ideal $G$ of the ring of differential polynomials $K[[t]]\{x_1,\ldots,x_n\}$, where $K$ is an uncountable algebraically closed field of characteristic zero. We define the tropicalization $\text{trop}(\text{Sol}(G))$ of the set of solutions $\text{Sol}(G)\subset K[[t]]^n$ of $G$, and the set of solutions associated to the tropicalization of the ideal $\text{trop}(G)$. These two sets are linked by a tropicalization morphism $\text{trop}:\text{Sol}(G)\longrightarrow \text{Sol}(\text{trop}(G))$. We show the equality $\text{trop}(\text{Sol}(G))=\text{Sol}(\text{trop}(G))$, answering a question raised by D. Grigoriev earlier this year.

Derivations On a C∞-ring

The main purpose of this paper is to provide several results on objects lying between differential geometry and algebraic geometry such as C∞-rings and derivations on a C∞-ring. A C∞-ring is defined as a set with operations by C∞-functions on Euclidean spaces. A derivation on a C∞-ring is defined in two method, an ℝ-derivation and a C∞-derivation. The main result of this paper is to show that any ℝ-derivation is a C∞-derivation for some classes of C∞-rings (Theorems 3.1, 3.2).

New effective differential Nullstellensatz

We show new upper and lower bounds for the effective differential Nullstellensatz for differential fields of characteristic zero with several commuting derivations. Seidenberg was the first to address this problem in 1956, without giving a complete solution. In the case of one derivation, the first bound is due to Grigoriev in 1989. The first bounds in the general case appeared in 2009 in a paper by Golubitsky, Kondratieva, Szanto, and Ovchinnikov, with the upper bound expressed in terms of the Ackermann function. D'Alfonso, Jeronimo, and Solernó, using novel ideas, obtained in 2014 a new bound if restricted to the case of one derivation and constant coefficients. To obtain the bound in the present paper without this restriction, we extend this approach and use the new methods of Freitag and León Sánchez and of Pierce, which represent a model-theoretic approach to differential algebraic geometry.

On bounds for the effective differential Nullstellensatz

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.

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.