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 (see Stokes Parameters) which has deep connections to the theory of D-Modules (see D-module) 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).

A biring $Q$ is a ring such that the functor it represents is a functor to the category of rings, so the affine scheme $\operatorname {Spec} Q$ is a ring scheme in the same way that the Spec of a of a Hopf algebra is a Group scheme (see Spectrum of a Ring).

Given a biring $Q$ and a ring $R$ an action can be viewed as a ring homomorphism $Q\odot R\to R$ or, equivalently, by adjunction, as a ring morphisms $R\to \operatorname {CRing} (Q,R)$. The functor $A\mapsto Q\odot A$ corresponds to taking jet bundles geometrically.

In the case of derivatives the ring scheme is $R\mapsto R[t]/(t^{2})$, the dual numbers functor. In the case of difference operations the ring scheme is the map $R\mapsto R\oplus R$. In the case of $p$-derivations the ring scheme is $R\mapsto W_{1}(R)$ the length two Witt vectors. Under this correspondence the comonad, $R\mapsto R[[t]]$, the ring of formal power series corresponds to the full ring of $p$-typical Witt vectors.

Basic Formulas

Formulas from calculus like the product rule and quotient rule apply in this setting.

In the following identities, $\delta$ is a derivation of a differential ring $R.$

• If $r\in R$ and $c$ is a constant in $R$ (that is, $\delta c=0$), then
  $$\delta (cr)=c\delta (r).$$
• If $r\in R$ and $u$ is a unit in $R,$ then
  $$\delta \left({\frac {r}{u}}\right)={\frac {\delta (r)u-r\delta (u)}{u^{2}}}$$
• If $n$ is a nonnegative integer and $r\in R$ then
  $$\delta (r^{n})=nr^{n-1}\delta (r)$$
• If $u_{1},\ldots ,u_{n}$ are units in $R,$ and $e_{1},\ldots ,e_{n}$ are integers, one has the logarithmic derivative identity:
  $${\frac {\delta (u_{1}^{e_{1}}\ldots u_{n}^{e_{n}})}{u_{1}^{e_{1}}\ldots u_{n}^{e_{n}}}}=e_{1}{\frac {\delta (u_{1})}{u_{1}}}+\dots +e_{n}{\frac {\delta (u_{n})}{u_{n}}}.$$

Higher-order derivations

A derivation operator or higher-order derivation is the composition of several derivations. As the derivations of a differential ring are supposed to commute, the order of the derivations does not matter, and a derivation operator may be written as

$$\delta _{1}^{e_{1}}\circ \cdots \circ \delta _{n}^{e_{n}},$$

where $\delta _{1},\ldots ,\delta _{n}$ are the derivations under consideration, $e_{1},\ldots ,e_{n}$ are nonnegative integers, and the exponent of a derivation denotes the number of times this derivation is composed in the operator.

The sum $o=e_{1}+\cdots +e_{n}$ is called the order of derivation. If $o=1$ the derivation operator is one of the original derivations. If $o=0$, one has the identity function, which is generally considered as the unique derivation operator of order zero. With these conventions, the derivation operators form a free commutative monoid on the set of derivations under consideration.

A derivative of an element $x$ of a differential ring is the application of a derivation operator to $x,$ that is, with the above notation, $\delta _{1}^{e_{1}}\circ \cdots \circ \delta _{n}^{e_{n}}(x).$ A proper derivative is a derivative of positive order.

Differential ideals

A differential ideal $I$ of a differential ring $R$ is an ideal of the ring $R$ such that ${\textstyle \partial x\in I,}$ for every derivation $\partial$ and every $x\in I.$

Algebraists often work with differential ideals that are radical --- radical ideals are sometimes called perfect in the differential algebra literature. A prime differential ideal is a differential ideal that is a prime in the usual sense.

A discovery of Ritt is that, although the classical theory of algebraic ideals does not work for differential ideals, a large part of it can be extended to radical differential ideals, and this makes them fundamental in differential algebra.

The intersection of any family of differential ideals is a differential ideal, and the intersection of any family of radical differential ideals is a radical differential ideal. It follows that, given a subset $S$ of a differential ring, there are three ideals generated by it, which are the intersections of, respectively, all algebraic ideals, all differential ideals, and all radical differential ideals that contain it.

The algebraic ideal generated by $S$ is the set of the finite linear combinations of elements of $S,$ and is commonly denoted as $(S)$ or $\langle S\rangle .$

The differential ideal generated by $S$ is the set of the finite linear combinations of elements of $S$ and of the derivatives of any order of these elements; it is commonly denoted as $[S].$ When $S$ is finite, $[S]$ is generally not finitely generated as an algebraic ideal.

The radical differential ideal generated by $S$ is commonly denoted as $\{S\}.$ There is no known way to characterize its element in a similar way as for the two other cases.

Differential polynomials

Differrential polynomials are just polynomials with derivatives in them. Let $(R,\partial )$ be an ordinary differential ring. The ring of differential polynomials in a single differential variable is the infinite polynomial ring $R\lbrace x\rbrace$ given by

$R\lbrace x\rbrace =R[x,x',x'',x''',\ldots ]$

where formal derivatives are adjoined to prolong the derivation $\partial$ on $R$ to the whole ring $R\lbrace x\rbrace$.

An element of this ring is called a differential polynomial. For example $u=(x'')^{2}(x')+x\in \mathbb {C} \lbrace x\rbrace$ is a differential polynomial. And we would have $\partial (u)=2x'x''x'''+(x'')^{3}+x'\in R\lbrace x\rbrace$.

When there are several derivatives $\partial _{i}$ and several variables $x_{1},\ldots ,x_{m}$ we form the commutative monoid $\Theta$ generated by the $\partial _{i}$ and then take the ring generated by the formal indeterminates $\theta (x_{j})$ for $\theta \in \Theta$. So we have

$R\lbrace x_{1},\ldots ,x_{n}\rbrace _{\Delta }=R[\partial ^{\alpha }(x_{j}):1\leq j\leq n,\alpha \in \mathbb {Z} _{\geq 0}^{m}]$

where in the above expression we have used multi-index notation where $\partial =(\partial _{1},\ldots ,\partial _{m})$, $\alpha =(\alpha _{1},\ldots ,\alpha _{m})$ and $\partial ^{\alpha }=\partial _{1}^{\alpha _{1}}\cdots \partial _{m}^{\alpha _{m}}$. So, for example, if $(\partial _{1},\partial _{2})=(\partial _{t},\partial _{x})$ and the differential variable is $u$ then $\partial _{t}(u)-6u\partial _{x}(u)-\partial _{x}^{3}(u)$ is a differential polynomial.

Independence of Solutions of Differential Equations

The notion of "transcendent" as in "Painleve transcendent" is covered by the notion of differential algebraic independence.

Let $(K,\partial )$ be a differential field. Let $(F,\partial )$ be a differential field extending $K$ (a common abuse is to just denote all derivatives by the same letter since it makes the formulas easier). We say that $\varphi \in K$ is "differential algebraic" over $F$ provided there exists some $f\in K\lbrace x\rbrace$ not contained in $K$ such that $f(\varphi )=0$.

For example, $\varphi =\sin(t)$ is differential algebraic over $K=\mathbb {Q}$ since it satisfies $f=x+x''\in K\lbrace x\rbrace .$

An element which is not differentially algebraic is called "differentially transcendental" (or "hypertranscendental" or "transcendent"). A famous theory of H\"{o}lder is that the Gamma function is hypertranscendental.

There are also famous theorems about the Painleve transcendents by Nishioka, Umemura and others.

The Galois theory of linear differential equations is called Picard-Vessiot Theory.

The Seidenberg Embedding Theorem says that every countable differential algebraically closed field (in one variable) can be embedding into a ring of germs of meromorphic functions.

Analog of the Hilbert Basis Theorem

If the field $K$ contains the field of rational numbers, the rings of differential polynomials over $K$ satisfy the ascending chain condition on radical differential ideals. This Ritt’s theorem is implied by its generalization, sometimes called the Ritt-Raudenbush basis theorem which asserts that if $R$ is a Ritt Algebra (that, is a differential ring containing the field of rational numbers), that satisfies the ascending chain condition on radical differential ideals, then the ring of differential polynomials $R\{y\}$ satisfies the same property (one passes from the univariate to the multivariate case by applying the theorem iteratively).

This Noetherian property implies that, in a ring of differential polynomials, every radical differential ideal I is finitely generated as a radical differential ideal; this means that there exists a finite set S of differential polynomials such that I is the smallest radical differential idesl containing S. This allows representing a radical differential ideal by such a finite set of generators, and computing with these ideals. However, some usual computations of the algebraic case cannot be extended. In particular no algorithm is known for testing membership of an element in a radical differential ideal or the equality of two radical differential ideals.

Another consequence of the Noetherian property is that a radical differential ideal can be uniquely expressed as the intersection of a finite number of prime differential ideals, called essential prime components of the ideal.

Algebras with derivations

Differential graded vector space

A ${\textstyle \operatorname {\mathbb {Z} -graded} }$ vector space ${\textstyle V_{\bullet }}$ is a collection of vector spaces ${\textstyle V_{m}}$ with integer degree ${\textstyle |v|=m}$ for ${\textstyle v\in V_{m}}$. A direct sum can represent this graded vector space:

$V_{\bullet }=\bigoplus _{m\in \mathbb {Z} }V_{m}$

A differential graded vector space or chain complex, is a graded vector space ${\textstyle V_{\bullet }}$ with a differential map or boundary map ${\textstyle d_{m}:V_{m}\to V_{m-1}}$ with $d_{m}\circ d_{m+1}=0$ .

A cochain complex is a graded vector space ${\textstyle V^{\bullet }}$ with a differential map or coboundary map ${\textstyle d_{m}:V_{m}\to V_{m+1}}$ with $d_{m+1}\circ d_{m}=0$.

Differential graded algebra

A differential graded algebra is a graded algebra ${\textstyle A}$ with a linear derivation ${\textstyle d:A\to A}$ with $d\circ d=0$ that follows the graded Leibniz product rule.

• Graded Leibniz product rule: $\forall a,b\in A,\ d(a\cdot b)=d(a)\cdot b+(-1)^{|a|}\cdot a\cdot d(b)$ with $|a|$ the degree of vector $a$.

Lie algebra

A Lie algebra is a finite-dimensional real or complex vector space ${\textstyle {\mathcal {g}}}$ with a bilinear bracket operator ${\textstyle [,]:{\mathcal {g}}\times {\mathcal {g}}\to {\mathcal {g}}}$ with Skew symmetry and the Jacobi identity property.

• Skew symmetry: $[X,Y]=-[Y,X]$
• Jacobi identity property: $[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0$

for all $X,Y,Z\in {\mathcal {g}}$.

The adjoint operator, ${\textstyle \operatorname {ad_{X}} (Y)=[Y,X]}$ is a derivation of the bracket because the adjoint's effect on the binary bracket operation is analogous to the derivation's effect on the binary product operation. This is the inner derivation determined by ${\textstyle X}$.

$\operatorname {ad} _{X}([Y,Z])=[\operatorname {ad} _{X}(Y),Z]+[Y,\operatorname {ad} _{X}(Z)]$

The universal enveloping algebra ${\textstyle U({\mathcal {g}})}$ of Lie algebra ${\textstyle {\mathcal {g}}}$ is a maximal associative algebra with identity, generated by Lie algebra elements ${\textstyle {\mathcal {g}}}$ and containing products defined by the bracket operation. Maximal means that a linear homomorphism maps the universal algebra to any other algebra that otherwise has these properties. The adjoint operator is a derivation following the Leibniz product rule.

• Product in $U({\mathcal {g}})$ : $X\cdot Y-Y\cdot X=[X,Y]$
• Leibniz product rule: $\operatorname {ad} _{X}(Y\cdot Z)=\operatorname {ad} _{X}(Y)\cdot Z+Y\cdot \operatorname {ad} _{X}(Z)$

for all $X,Y,Z\in U({\mathcal {g}})$.

Weyl algebras

One needs to exercise caution when looking at the literature on Weyl algebras because there are two inequivalent definitions that are present. Let $(R,\partial )$ be a differential ring. The Weyl algebra is the ring $R[\partial ]$ where the ring $\partial r=\partial (r)+r\partial$ described the commutation relation for $r\in R$.

In Noncommutative Geometry it is common to talk about the Weyl algebra as the special case where $(R,\partial )=(\mathbb {C} [x],\partial _{x})$ (or possibly in more partial derivatives). This, for example, is the Weyl algebra that appears in the Dixmier Conjecture.

In several variables the Weyl algebra is an algebra ${\textstyle A_{n}(K)=K[x_{1},\ldots ,x_{n},\partial _{x_{1}},\ldots ,\partial _{x_{n}}]}$ over a ring ${\textstyle K[p_{1},q_{1},\dots ,p_{n},q_{n}]}$ with a specific noncommutative product:

$p_{i}\cdot q_{i}-q_{i}\cdot p_{i}=1,\ :\ i\in \{1,\dots ,n\}$.

All other indeterminate products are commutative for ${\textstyle i,j\in \{1,\dots ,n\}}$:

$p_{i}\cdot q_{j}-q_{j}\cdot p_{i}=0{\text{ if }}i\neq j,\ p_{i}\cdot p_{j}-p_{j}\cdot p_{i}=0,\ q_{i}\cdot q_{j}-q_{j}\cdot q_{i}=0$.

A Weyl algebra can represent the derivations for a commutative ring's polynomials ${\textstyle f\in K[y_{1},\ldots ,y_{n}]}$. The Weyl algebra's elements are endomorphisms, the elements ${\textstyle p_{1},\ldots ,p_{n}}$ function as standard derivations, and map compositions generate linear differential operators. D-module is a related approach for understanding differential operators. The endomorphisms are:

$q_{j}(y_{k})=y_{j}\cdot y_{k},\ q_{j}(c)=c\cdot y_{j}{\text{ with }}c\in K,\ p_{j}(y_{j})=1,\ p_{j}(y_{k})=0{\text{ if }}j\neq k,\ p_{j}(c)=0{\text{ with }}c\in K$

Pseudodifferential Operators

The Weyl algebra $R[\partial ]$ is often completed to the ring $R[[\partial ^{-1}]]$ or $R((\partial ^{-1}))$ containing $R[\partial ]$ because it has nicer divisibility properties.

The associative, possibly noncommutative ring ${\textstyle A}$ has derivation ${\textstyle d:A\to A}$.

The pseudo-differential operator ring ${\textstyle A((\partial ^{-1}))}$ is a left ${\textstyle \operatorname {A-module} }$ containing ring elements ${\textstyle L}$:

$a_{i}\in A,\ i,i_{\min }\in \mathbb {N} ,\ |i_{\min }|>0\ :\ L=\sum _{i\geq i_{\min }}^{n}a_{i}\cdot \partial ^{i}$

The derivative operator is ${\textstyle d(a)=\partial \circ a-a\circ \partial }$.

The binomial coefficient is ${\Bigl (}{i \atop k}{\Bigr )}$.

Pseudo-differential operator multiplication is:

$\sum _{i\geq i_{\min }}^{n}a_{i}\cdot \partial ^{i}\cdot \sum _{j\geq j_{\min }}^{m}b_{i}\cdot \partial ^{j}=\sum _{i,j;k\geq 0}{\Bigl (}{i \atop k}{\Bigr )}\cdot a_{i}\cdot d^{k}(b_{j})\cdot \partial ^{i+j-k}$

Open problems

The Ritt problem asks is there an algorithm that determines if one prime differential ideal contains a second prime differential ideal when characteristic sets identify both ideals.

The Kolchin catenary conjecture states given a ${\textstyle d>0}$ dimensional irreducible differential algebraic variety ${\textstyle V}$ and an arbitrary point ${\textstyle p\in V}$, a long gap chain of irreducible differential algebraic subvarieties occurs from ${\textstyle p}$ to V.

The Jacobi bound conjecture concerns the upper bound for the order of an differential variety's irreducible component. The polynomial's orders determine a Jacobi number, and the conjecture is the Jacobi number determines this bound.{{sfn|Lando|1970|ps=none} This implies the Dimension Conjecture.

The Poincare Vector Field Problem: given polynomial vector field in the plane $a(x,y)\partial _{x}+b(x,y)\partial _{y}$ the question finding the invariant algebraic curves $f(x,y)=0$. This was stated at the same time as Hilbert's 16th Problem but remains open today. This was stated in 1885 and 1891. In Poincare's language: is it possible to recognize the genus of the general solution of an algebraic differential equation in two variables which has a rational first integral?

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 operatorsPages displaying wikidata descriptions as a fallback
• Hardy field – field consisting of germs of real-valued functions at infinity that are closed under differentiationPages displaying wikidata descriptions as a fallback
• 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

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• Stechlinski, Peter; Patrascu, Michael; Barton, Paul I. (2018). "Nonsmooth differential-algebraic equations in chemical engineering". Computers & Chemical Engineering. 114: 52–68. doi:10.1016/j.compchemeng.2017.10.031. hdl:1721.1/122980. S2CID 49413118.
• Taylor, Michael E. (1991). Pseudodifferential operators and nonlinear PDE. Boston: Birkhäuser. ISBN 978-0-8176-3595-4.
• Wu, Wen-tsün (2005a). "On "Good" Bases of Algebraic-Differential Ideals". Differential equations with symbolic computation. Birkhäuser. pp. 343–350. doi:10.1007/3-7643-7429-2_19. ISBN 978-3-7643-7429-7.
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• Zharinov, V. V. (December 2021). "Navier–Stokes equations, the algebraic aspect" (PDF). Theoretical and Mathematical Physics. 209 (3): 1657–1672. arXiv:2110.01504. Bibcode:2021TMP...209.1657Z. doi:10.1134/S0040577921120011. S2CID 238259977.

External links

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