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 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. Weyl algebras and Lie algebras may be considered as belonging to differential algebra.

More specifically, differential algebra refers to the theory introduced by Joseph Ritt in 1950, in which differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations. ^{[1] [2] [3]}

A natural example of a differential field is the field of rational functions in one variable over the complex numbers, ${\displaystyle \mathbb {C} (t),}$ where the derivation is differentiation with respect to ${\displaystyle t.}$ 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

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. ^[4] 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. ^{[5] [6] [2]} Ellis Kolchin, Ritt's student, advanced this field and published Differential Algebra And Algebraic Groups. ^[1]

Differential rings

Definition

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

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

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

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

A differential ring is a commutative ring ${\displaystyle R}$ equipped with one or more derivations that commute pairwise; that is, $${\displaystyle \partial _{1}(\partial _{2}(r))=\partial _{2}(\partial _{1}(r))}$$ for every pair of derivations and every ${\displaystyle r\in R.}$ ^[7] 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 a differential ring that is also a field. A differential algebra${\displaystyle A}$ over a differential field ${\displaystyle K}$ is a differential ring that contains ${\displaystyle K}$ as a subring such that the restriction to ${\displaystyle K}$ of the derivations of ${\displaystyle A}$ equal the derivations of ${\displaystyle K.}$ (A more general definition is given below, which covers the case where ${\displaystyle K}$ is not a field, and is essentially equivalent when ${\displaystyle K}$ is a field.)

A Witt algebra is a differential ring that contains the field ${\displaystyle \mathbb {Q} }$ of the rational numbers. Equivalently, this is a differential algebra over ${\displaystyle \mathbb {Q} ,}$ since ${\displaystyle \mathbb {Q} }$ can be considered as a differential field on which every derivation is the zero function.

The constants of a differential ring are the elements ${\displaystyle r}$ such that ${\displaystyle \partial r=0}$ for every derivation ${\displaystyle \partial .}$ The constants of a differential ring form a subring and the constants of a differentiable field form a subfield. ^[8] This meaning of "constant" generalizes the concept of a constant function, and must not be confused with the common meaning of a constant.

Basic formulas

In the following identities, ${\displaystyle \delta }$ is a derivation of a differential ring ${\displaystyle R.}$ ^[9]

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

Higher-order derivations

A derivation operator or higher-order derivation^{[ citation needed ]} 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 $${\displaystyle \delta _{1}^{e_{1}}\circ \cdots \circ \delta _{n}^{e_{n}},}$$ where ${\displaystyle \delta _{1},\ldots ,\delta _{n}}$ are the derivations under consideration, ${\displaystyle e_{1},\ldots ,e_{n}}$ are nonnegative integers, and the exponent of a derivation denotes the number of times this derivation is composed in the operator.

The sum ${\displaystyle o=e_{1}+\cdots +e_{n}}$ is called the order of derivation. If ${\displaystyle o=1}$ the derivation operator is one of the original derivations. If ${\displaystyle o=0}$, 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 ${\displaystyle x}$ of a differential ring is the application of a derivation operator to ${\displaystyle x,}$ that is, with the above notation, ${\displaystyle \delta _{1}^{e_{1}}\circ \cdots \circ \delta _{n}^{e_{n}}(x).}$ A proper derivative is a derivative of positive order. ^[7]

Differential ideals

A differential ideal ${\displaystyle I}$ of a differential ring ${\displaystyle R}$ is an ideal of the ring ${\displaystyle R}$ that is closed (stable) under the derivations of the ring; that is, ${\textstyle \partial x\in I,}$ for every derivation ${\displaystyle \partial }$ and every ${\displaystyle x\in I.}$ A differential ideal is said to be proper if it is not the whole ring. For avoiding confusion, an ideal that is not a differential ideal is sometimes called an algebraic ideal.

The radical of a differential ideal is the same as its radical as an algebraic ideal, that is, the set of the ring elements that have a power in the ideal. The radical of a differential ideal is also a differential ideal. A radical or perfect differential ideal is a differential ideal that equals its radical. ^[10] A prime differential ideal is a differential ideal that is prime in the usual sense; that is, if a product belongs to the ideal, at least one of the factors belongs to the ideal. A prime differential ideal is always a radical differential ideal.

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. ^[11] It follows that, given a subset ${\displaystyle S}$ 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. ^{[11] [12]}

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

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

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

Differential polynomials

A differential polynomial over a differential field ${\displaystyle K}$ is a formalization of the concept of differential equation such that the known functions appearing in the equation belong to ${\displaystyle K,}$ and the indeterminates are symbols for the unknown functions.

So, let ${\displaystyle K}$ be a differential field, which is typically (but not necessarily) a field of rational fractions ${\displaystyle K(X)=K(x_{1},\ldots ,x_{n})}$ (fractions of multivariate polynomials), equipped with derivations ${\displaystyle \partial _{i}}$ such that ${\displaystyle \partial _{i}x_{i}=1}$ and ${\displaystyle \partial _{i}x_{j}=0}$ if ${\displaystyle i\neq j}$ (the usual partial derivatives).

For defining the ring ${\textstyle K\{Y\}=K\{y_{1},\ldots ,y_{n}\}}$ of differential polynomials over ${\displaystyle K}$ with indeterminates in ${\displaystyle Y=\{y_{1},\ldots ,y_{n}\}}$ with derivations ${\displaystyle \partial _{1},\ldots ,\partial _{n},}$ one introduces an infinity of new indeterminates of the form ${\displaystyle \Delta y_{i},}$ where ${\displaystyle \Delta }$ is any derivation operator of order higher than 1. With this notation, ${\displaystyle K\{Y\}}$ is the set of polynomials in all these indeterminates, with the natural derivations (each polynomial involves only a finite number of indeterminates). In particular, if ${\displaystyle n=1,}$ one has

${\displaystyle K\{y\}=K\left[y,\partial y,\partial ^{2}y,\partial ^{3}y,\ldots \right].}$

Even when ${\displaystyle n=1,}$ a ring of differential polynomials is not Noetherian. This makes the theory of this generalization of polynomial rings difficult. However, two facts allow such a generalization.

Firstly, a finite number of differential polynomials involves together a finite number of indeterminates. Its follows that every property of polynomials that involves a finite number of polynomials remains true for differential polynomials. In particular, greatest common divisors exist, and a ring of differential polynomials is a unique factorization domain.

The second fact is that, if the field ${\displaystyle K}$ contains the field of rational numbers, the rings of differential polynomials over ${\displaystyle K}$ 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 ${\displaystyle R}$ is a Ritt Algebra (that, is a differential ring containing the field of rational numbers), ^[13] that satisfies the ascending chain condition on radical differential ideals, then the ring of differential polynomials ${\displaystyle R\{y\}}$ satisfies the same property (one passes from the univariate to the multivariate case by applying the theorem iteratively). ^{[14] [15]}

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. ^[16] 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. ^[17]

Elimination methods

Elimination methods are algorithms that preferentially eliminate a specified set of derivatives from a set of differential equations, commonly done to better understand and solve sets of differential equations.

Categories of elimination methods include characteristic set methods, differential Gröbner bases methods and resultant based methods. ^{[1] [18] [19] [20] [21] [22] [23]}

Common operations used in elimination algorithms include 1) ranking derivatives, polynomials, and polynomial sets, 2) identifying a polynomial's leading derivative, initial and separant, 3) polynomial reduction, and 4) creating special polynomial sets.

Ranking derivatives

The ranking of derivatives is a total order and an admisible order, defined as: ^{[24] [25] [26]}

${\textstyle \forall p\in \Theta Y,\ \forall \theta _{\mu }\in \Theta$ :\theta _{\mu }p>p.}

${\textstyle \forall p,q\in \Theta Y,\ \forall \theta _{\mu }\in \Theta :p\geq q\Rightarrow \theta _{\mu }p\geq \theta _{\mu }q.}$

Each derivative has an integer tuple, and a monomial order ranks the derivative by ranking the derivative's integer tuple. The integer tuple identifies the differential indeterminate, the derivative's multi-index and may identify the derivative's order. Types of ranking include: ^[27]

• Orderly ranking: ${\displaystyle \forall y_{i},y_{j}\in Y,\ \forall \theta _{\mu },\theta _{\nu }\in \Theta \$ :\ \operatorname {ord} (\theta _{\mu })\geq \operatorname {ord} (\theta _{\nu })\Rightarrow \theta _{\mu }y_{i}\geq \theta _{\nu }y_{j}}
• Elimination ranking: ${\displaystyle \forall y_{i},y_{j}\in Y,\ \forall \theta _{\mu },\theta _{\nu }\in \Theta \$ :\ y_{i}\geq y_{j}\Rightarrow \theta _{\mu }y_{i}\geq \theta _{\nu }y_{j}}

In this example, the integer tuple identifies the differential indeterminate and derivative's multi-index, and lexicographic monomial order, ${\textstyle \geq _{\text{lex}}}$, determines the derivative's rank. ^[28]

${\displaystyle \eta (\delta _{1}^{e_{1}}\circ \cdots \circ \delta _{n}^{e_{n}}(y_{j}))=(j,e_{1},\ldots ,e_{n})}$.

${\displaystyle \eta (\theta _{\mu }y_{j})\geq _{\text{lex}}\eta (\theta _{\nu }y_{k})\Rightarrow \theta _{\mu }y_{j}\geq \theta _{\nu }y_{k}.}$

Leading derivative, initial and separant

This is the standard polynomial form: ${\displaystyle p=a_{d}\cdot u_{p}^{d}+a_{d-1}\cdot u_{p}^{d-1}+\cdots +a_{1}\cdot u_{p}+a_{0}}$. ^{[24] [28]}

• Leader or leading derivative is the polynomial's highest ranked derivative: ${\displaystyle u_{p}}$.
• Coefficients ${\displaystyle a_{d},\ldots ,a_{0}}$ do not contain the leading derivative ${\textstyle u_{p}}$.
• Degree of polynomial is the leading derivative's greatest exponent: ${\displaystyle \deg _{u_{p}}(p)=d}$.
• Initial is the coefficient: ${\displaystyle I_{p}=a_{d}}$.
• Rank is the leading derivative raised to the polynomial's degree: ${\displaystyle u_{p}^{d}}$.
• Separant is the derivative: ${\displaystyle S_{p}={\frac {\partial p}{\partial u_{p}}}}$.

Separant set is ${\displaystyle S_{A}=\{S_{p}\mid p\in A\}}$, initial set is ${\displaystyle I_{A}=\{I_{p}\mid p\in A\}}$ and combined set is ${\textstyle H_{A}=S_{A}\cup I_{A}}$. ^[29]

Reduction

Partially reduced (partial normal form) polynomial ${\textstyle q}$ with respect to polynomial ${\textstyle p}$ indicates these polynomials are non-ground field elements, ${\textstyle p,q\in {\mathcal {K}}\{Y\}\setminus {\mathcal {K}}}$, and ${\displaystyle q}$ contains no proper derivative of ${\displaystyle u_{p}}$. ^{[30] [31] [29]}

Partially reduced polynomial ${\textstyle q}$ with respect to polynomial ${\textstyle p}$ becomes reduced (normal form) polynomial ${\textstyle q}$ with respect to ${\textstyle p}$ if the degree of ${\textstyle u_{p}}$ in ${\textstyle q}$ is less than the degree of ${\textstyle u_{p}}$ in ${\textstyle p}$. ^{[30] [31] [29]}

An autoreduced polynomial set has every polynomial reduced with respect to every other polynomial of the set. Every autoreduced set is finite. An autoreduced set is triangular meaning each polynomial element has a distinct leading derivative. ^{[32] [30]}

Ritt's reduction algorithm identifies integers ${\textstyle i_{A_{k}},s_{A_{k}}}$ and transforms a differential polynomial ${\textstyle f}$ using pseudodivision to a lower or equally ranked remainder polynomial ${\textstyle f_{red}}$ that is reduced with respect to the autoreduced polynomial set ${\textstyle A}$. The algorithm's first step partially reduces the input polynomial and the algorithm's second step fully reduces the polynomial. The formula for reduction is: ^[30]

${\displaystyle f_{\text{red}}\equiv \prod _{A_{k}\in A}I_{A_{k}}^{i_{A_{k}}}\cdot S_{A_{k}}^{i_{A_{k}}}\cdot f,{\pmod {[A]}}{\text{ with }}i_{A_{k}},s_{A_{k}}\in \mathbb {N} .}$

Ranking polynomial sets

Set ${\textstyle A}$ is a differential chain if the rank of the leading derivatives is ${\textstyle u_{A_{1}}<\dots <u_{A_{m}}}$ and ${\textstyle \forall i,\ A_{i}}$ is reduced with respect to ${\textstyle A_{i+1}}$ ^[33]

Autoreduced sets ${\textstyle A}$ and ${\textstyle B}$ each contain ranked polynomial elements. This procedure ranks two autoreduced sets by comparing pairs of identically indexed polynomials from both autoreduced sets. ^[34]

• ${\displaystyle A_{1}<\cdots <A_{m}\in A}$ and ${\displaystyle B_{1}<\cdots <B_{n}\in B}$ and ${\displaystyle i,j,k\in \mathbb {N} }$.
• ${\displaystyle {\text{rank }}A<{\text{rank }}B}$ if there is a ${\displaystyle k\leq \operatorname {minimum} (m,n)}$ such that ${\displaystyle A_{i}=B_{i}}$ for ${\textstyle 1\leq i<k}$ and ${\displaystyle A_{k}<B_{k}}$.
• ${\displaystyle \operatorname {rank} A<\operatorname {rank} B}$ if ${\displaystyle n<m}$ and ${\displaystyle A_{i}=B_{i}}$ for ${\displaystyle 1\leq i\leq n}$.
• ${\displaystyle \operatorname {rank} A=\operatorname {rank} B}$ if ${\displaystyle n=m}$ and ${\displaystyle A_{i}=B_{i}}$ for ${\displaystyle 1\leq i\leq n}$.

Polynomial sets

A characteristic set${\textstyle C}$ is the lowest ranked autoreduced subset among all the ideal's autoreduced subsets whose subset polynomial separants are non-members of the ideal ${\textstyle {\mathcal {I}}}$. ^[35]

The delta polynomial applies to polynomial pair ${\textstyle p,q}$ whose leaders share a common derivative, ${\textstyle \theta _{\alpha }u_{p}=\theta _{\beta }u_{q}}$. The least common derivative operator for the polynomial pair's leading derivatives is ${\textstyle \theta _{pq}}$, and the delta polynomial is: ^{[36] [37]}

${\displaystyle \operatorname {\Delta -poly} (p,q)=S_{q}\cdot {\frac {\theta _{pq}p}{\theta _{p}}}-S_{p}\cdot {\frac {\theta _{pq}q}{\theta _{q}}}}$

A coherent set is a polynomial set that reduces its delta polynomial pairs to zero. ^{[36] [37]}

Regular system and regular ideal

A regular system${\textstyle \Omega }$ contains a autoreduced and coherent set of differential equations ${\textstyle A}$ and a inequation set ${\textstyle H_{\Omega }\supseteq H_{A}}$ with set ${\textstyle H_{\Omega }}$ reduced with respect to the equation set. ^[37]

Regular differential ideal ${\textstyle {\mathcal {I}}_{\text{dif}}}$ and regular algebraic ideal ${\textstyle {\mathcal {I}}_{\text{alg}}}$ are saturation ideals that arise from a regular system. ^[37] Lazard's lemma states that the regular differential and regular algebraic ideals are radical ideals. ^[38]

• Regular differential ideal: ${\textstyle {\mathcal {I}}_{\text{dif}}=[A]:H_{\Omega }^{\infty }.}$
• Regular algebraic ideal: ${\textstyle {\mathcal {I}}_{\text{alg}}=(A):H_{\Omega }^{\infty }.}$

Rosenfeld–Gröbner algorithm

The Rosenfeld–Gröbner algorithm decomposes the radical differential ideal as a finite intersection of regular radical differential ideals. These regular differential radical ideals, represented by characteristic sets, are not necessarily prime ideals and the representation is not necessarily minimal. ^[39]

The membership problem is to determine if a differential polynomial ${\textstyle p}$ is a member of an ideal generated from a set of differential polynomials ${\textstyle S}$. The Rosenfeld–Gröbner algorithm generates sets of Gröbner bases. The algorithm determines that a polynomial is a member of the ideal if and only if the partially reduced remainder polynomial is a member of the algebraic ideal generated by the Gröbner bases. ^[40]

The Rosenfeld–Gröbner algorithm facilitates creating Taylor series expansions of solutions to the differential equations. ^[41]

Examples

Differential fields

Example 1: ${\textstyle (\operatorname {Mer} (\operatorname {f} (y),\partial _{y}))}$ is the differential meromorphic function field with a single standard derivation.

Example 2: ${\textstyle (\mathbb {C} \{y\},p(y)\cdot \partial _{y})}$ is a differential field with a linear differential operator as the derivation, for any polynomial ${\displaystyle p(y)}$.

Derivation

Define ${\textstyle E^{a}(p(y))=p(y+a)}$ as shift operator ${\textstyle E^{a}}$ for polynomial ${\textstyle p(y)}$.

A shift-invariant operator ${\textstyle T}$ commutes with the shift operator: ${\textstyle E^{a}\circ T=T\circ E^{a}}$.

The Pincherle derivative , a derivation of shift-invariant operator ${\textstyle T}$, is ${\textstyle T^{\prime }=T\circ y-y\circ T}$. ^[42]

Constants

Ring of integers is ${\displaystyle (\mathbb {Z} .\delta )}$, and every integer is a constant.

• The derivation of 1 is zero. ${\textstyle \delta (1)=\delta (1\cdot 1)=\delta (1)\cdot 1+1\cdot \delta (1)=2\cdot \delta (1)\Rightarrow \delta (1)=0}$.
• Also, ${\displaystyle \delta (m+1)=\delta (m)+\delta (1)=\delta (m)\Rightarrow \delta (m+1)=\delta (m)}$.
• By induction, ${\displaystyle \delta (1)=0\ \wedge \ \delta (m+1)=\delta (m)\Rightarrow \forall \ m\in \mathbb {Z} ,\ \delta (m)=0}$.

Field of rational numbers is ${\displaystyle (\mathbb {Q} .\delta )}$, and every rational number is a constant.

• Every rational number is a quotient of integers.

  ${\displaystyle \forall r\in \mathbb {Q} ,\ \exists \ a\in \mathbb {Z} ,\ b\in \mathbb {Z} /\{0\},\ r={\frac {a}{b}}}$

• Apply the derivation formula for quotients recognizing that derivations of integers are zero:

  ${\displaystyle \delta (r)=\delta \left({\frac {a}{b}}\right)={\frac {\delta (a)\cdot b-a\cdot \delta (b)}{b^{2}}}=0}$.

Differential subring

Constants form the subring of constants${\textstyle (\mathbb {C} ,\partial _{y})\subset (\mathbb {C} \{y\},\partial _{y})}$. ^[43]

Differential ideal

Element ${\textstyle \exp(y)}$ simply generates differential ideal ${\textstyle [\exp(y)]}$ in the differential ring ${\textstyle (\mathbb {C} \{y,\exp(y)\},\partial _{y})}$. ^[44]

Algebra over a differential ring

Any ring with identity is a ${\textstyle \operatorname {{\mathcal {Z}}-} }$algebra. ^[45] Thus a differential ring is a ${\textstyle \operatorname {{\mathcal {Z}}-} }$algebra.

If ring ${\textstyle {\mathcal {R}}}$ is a subring of the center of unital ring ${\textstyle {\mathcal {M}}}$, then ${\textstyle {\mathcal {M}}}$ is an ${\textstyle \operatorname {{\mathcal {R}}-} }$algebra. ^[45] Thus, a differential ring is an algebra over its differential subring. This is the natural structure of an algebra over its subring. ^[30]

Special and normal polynomials

Ring ${\textstyle (\mathbb {Q} \{y,z\},\partial _{y})}$ has irreducible polynomials, ${\textstyle p}$ (normal, squarefree) and ${\textstyle q}$ (special, ideal generator).

${\textstyle \partial _{y}(y)=1,\ \partial _{y}(z)=1+z^{2},\ z=\tan(y)}$

${\textstyle p(y)=1+y^{2},\ \partial _{y}(p)=2\cdot y,\ \gcd(p,\partial _{y}(p))=1}$

${\textstyle q(z)=1+z^{2},\ \partial _{y}(q)=2\cdot z\cdot (1+z^{2}),\ \gcd(q,\partial _{y}(q))=q}$

Polynomials

Ranking

Ring ${\textstyle (\mathbb {Q} \{y_{1},y_{2}\},\delta )}$ has derivatives ${\textstyle \delta (y_{1})=y_{1}^{\prime }}$ and ${\textstyle \delta (y_{2})=y_{2}^{\prime }}$

• Map each derivative to an integer tuple: ${\textstyle \eta (\delta ^{(i_{2})}(y_{i_{1}}))=(i_{1},i_{2})}$.
• Rank derivatives and integer tuples: ${\textstyle y_{2}^{\prime \prime }\ (2,2)>y_{2}^{\prime }\ (2,1)>y_{2}\ (2,0)>y_{1}^{\prime \prime }\ (1,2)>y_{1}^{\prime }\ (1,1)>y_{1}\ (1,0)}$.

Leading derivative and initial

The leading derivatives, and initials are:

${\textstyle p={\color {Blue}(y_{1}+y_{1}^{\prime })}\cdot ({\color {Red}y_{2}^{\prime \prime }})^{2}+3\cdot y_{1}^{2}\cdot {\color {Red}y_{2}^{\prime \prime }}+(y_{1}^{\prime })^{2}}$

${\textstyle q={\color {Blue}(y_{1}+3\cdot y_{1}^{\prime })}\cdot {\color {Red}y_{2}^{\prime \prime }}+y_{1}\cdot y_{2}^{\prime }+(y_{1}^{\prime })^{2}}$

${\textstyle r={\color {Blue}(y_{1}+3)}\cdot ({\color {Red}y_{1}^{\prime \prime }})^{2}+y_{1}^{2}\cdot {\color {Red}y_{1}^{\prime \prime }}+2\cdot y_{1}}$

Separants

${\textstyle S_{p}=2\cdot (y_{1}+y_{1}^{\prime })\cdot y_{2}^{\prime \prime }+3\cdot y_{1}^{2}}$.

${\textstyle S_{q}=y_{1}+3\cdot y_{1}^{\prime }}$

${\textstyle S_{r}=2\cdot (y_{1}+3)\cdot y_{1}^{\prime \prime }+y_{1}^{2}}$

Autoreduced sets

• Autoreduced sets are ${\textstyle \{p,r\}}$ and ${\textstyle \{q,r\}}$. Each set is triangular with a distinct polynomial leading derivative.
• The non-autoreduced set ${\textstyle \{p,q\}}$ contains only partially reduced ${\textstyle p}$ with respect to ${\textstyle q}$; this set is non-triangular because the polynomials have the same leading derivative.

Applications

Symbolic integration

Symbolic integration uses algorithms involving polynomials and their derivatives such as Hermite reduction, Czichowski algorithm, Lazard-Rioboo-Trager algorithm, Horowitz-Ostrogradsky algorithm, squarefree factorization and splitting factorization to special and normal polynomials. ^[46]

Differential equations

Differential algebra can determine if a set of differential polynomial equations has a solution. A total order ranking may identify algebraic constraints. An elimination ranking may determine if one or a selected group of independent variables can express the differential equations. Using triangular decomposition and elimination order, it may be possible to solve the differential equations one differential indeterminate at a time in a step-wise method. Another approach is to create a class of differential equations with a known solution form; matching a differential equation to its class identifies the equation's solution. Methods are available to facilitate the numerical integration of a differential-algebraic system of equations. ^[47]

In a study of non-linear dynamical systems with chaos, researchers used differential elimination to reduce differential equations to ordinary differential equations involving a single state variable. They were successful in most cases, and this facilitated developing approximate solutions, efficiently evaluating chaos, and constructing Lyapunov functions. ^[48] Researchers have applied differential elimination to understanding cellular biology, compartmental biochemical models, parameter estimation and quasi-steady state approximation (QSSA) for biochemical reactions. ^{[49] [50]} Using differential Gröbner bases, researchers have investigated non-classical symmetry properties of non-linear differential equations. ^[51] Other applications include control theory, model theory, and algebraic geometry. ^{[52] [16] [53]} Differential algebra also applies to differential-difference equations. ^[54]

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: ^[55]

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

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 ${\displaystyle d_{m}\circ d_{m+1}=0}$ . ^[56]

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 ${\displaystyle d_{m+1}\circ d_{m}=0}$. ^[56]

Differential graded algebra

A differential graded algebra is a graded algebra ${\textstyle A}$ with a linear derivation ${\textstyle d:A\to A}$ with ${\displaystyle d\circ d=0}$ that follows the graded Leibniz product rule. ^[57]

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

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. ^[58]

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

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

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 ${\textstyle X}$. ^{[59] [60]}

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

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. ^[61]

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

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

Weyl algebra

The Weyl algebra is an algebra ${\textstyle A_{n}(K)}$ over a ring ${\textstyle K[p_{1},q_{1},\dots ,p_{n},q_{n}]}$ with a specific noncommutative product: ^[62]

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

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

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

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: ^[62]

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

Pseudodifferential operator ring

The associative, possibly noncommutative ring ${\textstyle A}$ has derivation ${\textstyle d:A\to A}$. ^[63]

The pseudo-differential operator ring${\textstyle A((\partial ^{-1}))}$ is a left ${\textstyle \operatorname {A-module} }$ containing ring elements ${\textstyle L}$: ^{[63] [64] [65]}

${\displaystyle 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 ${\textstyle d(a)=\partial \circ a-a\circ \partial }$. ^[63]

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

Pseudo-differential operator multiplication is: ^[63]

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

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. ^[66]

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. ^[67]

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. ^[68]

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  – 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
• Kolchin's problems

Citations

1. Kolchin 1973
2. Ritt 1950
3. Kaplansky 1976
4. Ritt 1932, pp. iii–iv
5. Ritt 1930
6. Ritt 1932
7. Kolchin 1973, pp. 58–59
8. Kolchin 1973, pp. 58–60
9. Bronstein 2005, p. 76
10. Sit 2002, pp. 3–4
11. Kolchin 1973, pp. 61–62
12. Buium 1994, p. 21
13. Kaplansky 1976, p. 12
14. Kaplansky 1976, pp. 45, 48, 56–57
15. Kolchin 1973, pp. 126–129
16. Marker 2000
17. Hubert 2002, p. 8
18. Li & Yuan 2019
19. Boulier et al. 1995
20. Mansfield 1991
21. Ferro 2005
22. Chardin 1991
23. Wu 2005b
24. Kolchin 1973, pp. 75–76
25. Gao et al. 2009, p. 1141
26. Hubert 2002, p. 10
27. Ferro & Gerdt 2003, p. 83
28. Wu 2005a, p. 4
29. Boulier et al. 1995, p. 159
30. Kolchin 1973, p. 75
31. Ferro & Gerdt 2003, p. 84
32. Sit 2002, p. 6
33. Li & Yuan 2019, p. 294
34. Kolchin 1973, p. 81
35. Kolchin 1973, p. 82
36. Kolchin 1973, p. 136
37. Boulier et al. 1995, p. 160
38. Morrison 1999
39. Boulier et al. 1995, p. 158
40. Boulier et al. 1995, p. 164
41. Boulier et al. 2009b
42. Rota, Kahaner & Odlyzko 1973, p. 694
43. Kolchin 1973, p. 60
44. Sit 2002, p. 4
45. Dummit & Foote 2004, p. 343
46. Bronstein 2005, pp. 41, 51, 53, 102, 299, 309
47. Hubert 2002, pp. 41–47
48. Harrington & VanGorder 2017
49. Boulier 2007
50. Boulier & Lemaire 2009a
51. Clarkson & Mansfield 1994
52. Diop 1992
53. Buium 1994
54. Gao et al. 2009
55. Keller 2019, p. 48
56. Keller 2019, pp. 50–51
57. Keller 2019, pp. 58–59
58. Hall 2015, p. 49
59. Hall 2015, p. 51
60. Jacobson 1979, p. 9
61. Hall 2015, p. 247
62. Lam 1991, pp. 7–8
63. Parshin 1999, p. 268
64. Dummit & Foote 2004, p. 337
65. Taylor 1991
66. Golubitsky, Kondratieva & Ovchinnikov 2009
67. Freitag, Sánchez & Simmons 2016
68. Lando 1970

References

• Boulier, François; Lazard, Daniel; Ollivier, François; Petitot, Michel (1995). "Representation for the radical of a finitely generated differential ideal". Proceedings of the 1995 international symposium on Symbolic and algebraic computation – ISSAC '95 (PDF). pp. 158–166. doi:10.1145/220346.220367. ISBN   0897916999. S2CID   11059042.
• Boulier, François (31 December 2007). "Differential Elimination and Biological Modelling". Gröbner Bases in Symbolic Analysis. 2: 109–138. doi:10.1515/9783110922752.109. ISBN   978-3-11-019323-7. S2CID   61916692.
• Boulier, François; Lemaire, François (2009a). "Differential algebra and QSSA methods in biochemistry". IFAC Proceedings Volumes. 42 (10): 33–38. doi:10.3182/20090706-3-FR-2004.00004.
• Boulier, François; Lazard, Daniel; Ollivier, François; Petitot, Michel (April 2009b). "Computing representations for radicals of finitely generated differential ideals". Applicable Algebra in Engineering, Communication and Computing. 20 (1): 73–121. doi:10.1007/s00200-009-0091-7. S2CID   5482290.
• Bronstein, Manuel (2005). Symbolic integration I : transcendental functions. Algorithms and Computation in Mathematics. Vol. 1 (2nd ed.). Berlin: Springer. doi:10.1007/b138171. ISBN   3-540-21493-3.
• Buium, Alexandru (1994). Differential algebra and diophantine geometry. Hermann. ISBN   978-2-7056-6226-4.
• Chardin, Marc (1991). "Differential resultants and subresultants". In Budach, L. (ed.). Fundamentals of Computation Theory. FCT 1991. Lecture Notes in Computer Science. Vol. 529. Berlin, Heidelberg: Springer. pp. 180–189. doi:10.1007/3-540-54458-5_62. ISBN   978-3-540-38391-8.
• Clarkson, Peter A.; Mansfield, Elizabeth L. (January 1994). "Symmetry reductions and exact solutions of a class of nonlinear heat equations". Physica D: Nonlinear Phenomena. 70 (3): 250–288. arXiv: solv-int/9306002 . Bibcode:1994PhyD...70..250C. doi:10.1016/0167-2789(94)90017-5. S2CID   16858637.
• Crespo, Teresa; Hajto, Zbigniew (2011). Algebraic groups and differential Galois theory. Providence, R.I.: American Mathematical Society. ISBN   978-0-8218-5318-4.
• Diop, Sette (May 1992). "Differential-algebraic decision methods and some applications to system theory" (PDF). Theoretical Computer Science. 98 (1): 137–161. doi:10.1016/0304-3975(92)90384-R.
• Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons. ISBN   0-471-43334-9.
• Ferro, Giuseppa Carrá; Gerdt, V. P. (2003). "Improved Kolchin–Ritt Algorithm". Programming and Computer Software. 29 (2): 83–87. doi:10.1023/A:1022996615890. S2CID   26280002.
• Ferro, Giuseppa Carrá (2005). "Generalized Differential Resultant Systems of Algebraic ODEs and Differential Elimination Theory". Differential equations with symbolic computation. Trends in Mathematics. Birkhäuser. pp. 343–350. doi:10.1007/3-7643-7429-2_18. ISBN   978-3-7643-7429-7.
• Freitag, James; Sánchez, Omar León; Simmons, William (2 June 2016). "On Linear Dependence Over Complete Differential Algebraic Varieties". Communications in Algebra. 44 (6): 2645–2669. arXiv: 1401.6211 . doi:10.1080/00927872.2015.1057828. S2CID   56218725.
• Gao, X. S.; Van der Hoeven, J.; Yuan, C. M.; Zhang, G. L. (1 September 2009). "Characteristic set method for differential–difference polynomial systems". Journal of Symbolic Computation. 44 (9): 1137–1163. doi: 10.1016/j.jsc.2008.02.010 .
• Golubitsky, O. D.; Kondratieva, M. V.; Ovchinnikov, A. I. (2009). "On the generalized Ritt problem as a computational problem". Journal of Mathematical Sciences. 163 (5): 515–522. arXiv: 0809.1128 . doi:10.1007/s10958-009-9689-3. S2CID   17503904.
• Hall, Brian C. (2015). Lie groups, Lie algebras, and representations: an elementary introduction (Second ed.). Cham: Springer. ISBN   978-3-319-13467-3.
• Harrington, Heather A.; VanGorder, Robert A. (2017). "Reduction of dimension for nonlinear dynamical systems". Nonlinear Dynamics. 88 (1): 715–734. doi:10.1007/s11071-016-3272-5. PMC   7089670 . PMID   32226227. S2CID   254893812.
• Hubert, Evelyne (2002). "Notes on Triangular Sets and Triangulation-Decomposition Algorithms II: Differential Systems". In Winkler, Franz; Langer, Ulrich (eds.). Symbolic and Numerical Scientific Computing. Second International Conference, SNSC 2001 Hagenberg, Austria, September 12-14, 2001 Revised Papers (PDF). Berlin: Springer-Verlag. pp. 40–87. ISBN   3-540-40554-2.
• Jacobson, Nathan (1979). Lie algebras. New York. ISBN   0-486-63832-4.
• Kaplansky, Irving (1976). An introduction to differential algebra (2nd ed.). Hermann. ISBN   9782705612511.
• Keller, Corina (2019). Chern-Simons theory and equivariant factorization algebras. BestMasters. Wiesbaden, Germany. doi:10.1007/978-3-658-25338-7. ISBN   978-3-658-25337-0. S2CID   128325519.
• Kolchin, Ellis (1973). Differential Algebra And Algebraic Groups. Academic Press. ISBN   978-0-08-087369-5.
• Lam, T. Y. (1991). A first course in noncommutative rings. Graduate Texts in Mathematics. Vol. 131. New York: Springer-Verlag. doi:10.1007/978-1-4419-8616-0. ISBN   0-387-97523-3.
• Lando, Barbara A. (1970). "Jacobi's bound for the order of systems of first order differential equations". Transactions of the American Mathematical Society. 152 (1): 119–135. doi: 10.1090/S0002-9947-1970-0279079-1 . ISSN   0002-9947.
• Li, Wei; Yuan, Chun-Ming (February 2019). "Elimination Theory in Differential and Difference Algebra". Journal of Systems Science and Complexity. 32 (1): 287–316. doi:10.1007/s11424-019-8367-x. S2CID   255158214.
• Marker, David (2000). "Model theory of differential fields". In Haskell, Deirdre; Pillay, Anand; Steinhorn, Charles (eds.). Model theory, algebra, and geometry (PDF). Vol. 39. Cambridge: Cambridge University Press. pp. 53–64. ISBN   0-521-78068-3.
• Mansfield, Elizabeth (1991). Differential Bases (PhD). University of Sydney.
• Morrison, Sally (1 October 1999). "The Differential Ideal [ P ] : M∞" (PDF). Journal of Symbolic Computation. 28 (4): 631–656. doi:10.1006/jsco.1999.0318. ISSN   0747-7171.
• Parshin, Aleksei Nikolaevich (1999). "On a ring of formal pseudo-differential operators". Proc. Steklov Math. Institute. 224: 266–280. arXiv: math/9911098 . Bibcode:1999math.....11098P.
• Ritt, Joseph Fels (1930). "Manifolds of functions defined by systems of algebraic differential equations" (PDF). Transactions of the American Mathematical Society. 32 (4): 569–598. doi:10.1090/S0002-9947-1930-1501554-4. S2CID   54064812.
• Ritt, Joseph (1932). differential equations from the algebraic standpoint. Vol. 14. American Mathematical Society.
• Ritt, Joseph Fels (1950). Differential Algebra. Vol. 33. Providence, Rhode Island: American Mathematical Society Colloquium Publications. ISBN   978-0-8218-3205-9.
• Rota, Gian-Carlo; Kahaner, David; Odlyzko, Andrew (1973). "On the foundations of combinatorial theory. VIII. Finite operator calculus". Journal of Mathematical Analysis and Applications. 42 (3): 684–760. doi: 10.1016/0022-247X(73)90172-8 .
• Sit, William Y. (2002). "The Ritt-Kolchin theory for differential polynomials". In Guo, Li; Cassidy, Phyllis J; Keigher, William F; Sit, William Y (eds.). Differential algebra and related topics: proceedings of the International Workshop, Newark Campus of Rutgers, the State University of New Jersey, 2-3 November 2000. River Edge, NJ: World Scientific. doi:10.1142/4768. ISBN   981-02-4703-6.
• 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.
• Wu, Wen-tsün (2005b). "On the Construction of Groebner Basis of a Polynomial Ideal Based on Riquier–Janet Theory". Differential equations with symbolic computation. Trends in Mathematics. Birkhäuser. pp. 351–368. doi:10.1007/3-7643-7429-2_20. ISBN   978-3-7643-7429-7.
• 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.