Differential algebra

This article is about algebraic study of differential equations. For the concept in homological algebra, see Differential graded algebra.

In mathematics, differential algebra is, broadly speaking, the study of rings with extra operations includin. This for example includes difference algebra and wittferential algebra which is about p-derivations.

Classical, differential algebra, which this article will focus on, is area of mathematics consisting in the study of differential equations and differential operators as algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Differential algebra is to is to Differential Algebraic Geometry (the study of the geometry of algebraic differential equations) as Commutative Algebra is to Algebraic Geometry. The theory has close connections to the Theory of Holomorphic Foliations in the same way that Algebraic Geometry is closely related to Complex Geometry (the study of complex manifolds).

Let $\Delta =\lbrace \partial _{1},\ldots ,\partial _{m}\rbrace$ be a collection of commuting formal derivations on a ring $R$. That is, each $\partial \in \Delta$ satisfies a sum rule $\partial (f+g)=\partial (f)+\partial (g)$, product rule $\partial (fg)=\partial (f)g+f\partial (g)$, and $\partial (0)=0,\partial (1)=0$ .

Often writers restrict to the case of a single derivation when $\Delta =\lbrace \partial \rbrace$ and when these two cases need to be distinguished are referred to as "ordinary differential algebra" or "partial differential algebra" corresponding to the terminology of ordinary and partial differential equations.

A natural example of a differential field is the field of rational functions in one variable over the complex numbers, $\mathbb {C} (t),$ where the derivation is differentiation with respect to $t.$ More generally, every differential equation may be viewed as an element of a differential algebra over the differential field generated by the (known) functions appearing in the equation.

History

Many theorem from Differential algebra are quite old and go as far back as to Newton, Euler, Lagrange. Some of its modern problems date back to Poincare and Jacobi. For example the problem of when integrals can actually be integrated and why $\int e^{x^{2}}dx$ has no solution in "elementary functions" was investigated by Liouville. The galois theory for linear differential equations is closely tied to the monodromy group where fundamental matrices change after winding around a loop. In the case of irregular singularities this leads the the study of Stokes Phenomena which has deep connections to the theory of D-Modules developed by Bernard Malgrange and others.

Its modern form, which mirrors algebraic geometry, started with Joseph Ritt. Joseph Ritt developed differential algebra because he viewed attempts to reduce systems of differential equations to various canonical forms as an unsatisfactory approach. However, the success of algebraic elimination methods and algebraic manifold theory motivated Ritt to consider a similar approach for differential equations. His efforts led to an initial paper Manifolds Of Functions Defined By Systems Of Algebraic Differential Equations and 2 books, Differential Equations From The Algebraic Standpoint and Differential Algebra. Ellis Kolchin, Ritt's student, advanced this field and published Differential Algebra And Algebraic Groups.

Differential rings

Definition

A derivation ${\textstyle \partial }$ on a ring ${\textstyle R}$ is a function $\partial :R\to R\,$ such that $$\partial (r_{1}+r_{2})=\partial r_{1}+\partial r_{2}$$ and

$\partial (r_{1}r_{2})=(\partial r_{1})r_{2}+r_{1}(\partial r_{2})\quad$ (Leibniz product rule),

for every $r_{1}$ and $r_{2}$ in $R.$

A derivation is linear over the integers since these identities imply $\partial (0)=\partial (1)=0$ and $\partial (-r)=-\partial (r).$

A differential ring is a commutative ring $R$ equipped with one or more derivations that commute pairwise; that is, $$\partial _{1}(\partial _{2}(r))=\partial _{2}(\partial _{1}(r))$$ for every pair of derivations and every $r\in R.$ When there is only one derivation one talks often of an ordinary differential ring; otherwise, one talks of a partial differential ring.

A differential field is differentiable ring that is also a field. A differential algebra $A$ over a differential field $K$ is a differential ring that contains $K$ as a subring such that the restriction to $K$ of the derivations of $A$ equal the derivations of $K.$ (A more general definition is given below, which covers the case where $K$ is not a field, and is essentially equivalent when $K$ is a field.)

(A Ritt algebra is a differential ring that contains the field $\mathbb {Q}$ of the rational numbers. Equivalently, this is a differential algebra over $\mathbb {Q} ,$ since $\mathbb {Q}$ can be considered as a differential field on which every derivation is the zero function. This isn't a central concept but sometimes appears.)

The constants of a differential ring are the elements $r$ such that $\partial r=0$ for every derivation $\partial .$ The constants of a differential ring form a subring and the constants of a differentiable field form a subfield. This meaning of "constant" generalizes the concept of a constant function, and must not be confused with the common meaning of a constant.

Advanced Setup

Ring with general operations (e.g. difference rings or p-derivations) can be encoded with Bialgebras (usually called Birings in this literature). Tall, David O., and Gavin C. Wraith. "Representable functors and operations on rings." Proceedings of the London Mathematical Society 3.4 (1970): 619-643.

The classification of strongly minimal differential algebraic varieties. This is related to the Model Theory of Differential equations and Zariski Geometries.

Ferra Carro's Problem on the existence of differential Groebner bases.

The p-Curvature Conjecture of Katz and Grothendieck.

Higher order versions of the Painleve equations with no movable singularities.

See Kolchin's Problems.

See also

• Arithmetic derivative – Function defined on integers in number theory
• Difference algebra
• Differential algebraic geometry
• Differential calculus over commutative algebras – part of commutative algebraPages displaying wikidata descriptions as a fallback
• Differential Galois theory – Study of Galois symmetry groups of differential fields
• Differentially closed field
• Differential graded algebra – Algebraic structure in homological algebra
• D-module – Module over a sheaf of differential operators
• Hardy field – Mathematical concept
• Kähler differential – Differential form in commutative algebra
• Liouville's theorem (differential algebra) – Says when antiderivatives of elementary functions can be expressed as elementary functions
• Picard–Vessiot theory – Study of differential field extensions induced by linear differential equations

References

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External links

• David Marker's home page has several online surveys discussing differential fields.