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{{Short description|Algebraic study of differential equations}}
{{about|algebraic study of differential equations|the concept in homological algebra|Differential graded algebra}}

In [[mathematics]], '''differential algebra''' is, broadly speaking, the area of mathematics consisting in the study of [[differential equation]]s and [[differential operators]] as [[algebra|algebraic objects]] in view of deriving properties of differential equations and operators without computing the solutions,  similarly as [[polynomial algebra]]s are used for the study of [[algebraic varieties]], which are solution sets of [[systems of polynomial equations]]. [[Weyl algebra]]s and [[Lie algebra]]s 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 [[ring (mathematics)|rings]], [[field (mathematics)|fields]], and [[algebra over a field|algebras]] equipped with finitely many [[derivation (differential algebra)|derivations]].{{sfn|Kolchin|1973|ps=none}}{{sfn|Ritt|1950|ps=none}}{{sfn|Kaplansky|1976|ps=none}}

A natural example of a differential field is the field of [[rational function]]s in one variable over the [[complex number]]s, <math>\mathbb{C}(t),</math> where the derivation is differentiation with respect to <math>t.</math> 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.{{sfn|Ritt|1932|pp=iii–iv|ps=none}}  His efforts led to an initial paper <em>Manifolds Of Functions Defined By Systems Of Algebraic Differential Equations</em> and 2 books, <em>Differential Equations From The Algebraic Standpoint</em> and <em>Differential Algebra</em>.{{sfn|Ritt|1930|ps=none}}{{sfn|Ritt|1932|ps=none}}{{sfn|Ritt|1950|ps=none}}  [[Ellis Kolchin]], Ritt's student, advanced this field and published <em>Differential Algebra And Algebraic Groups</em>.{{sfn|Kolchin|1973|ps=none}}

== Differential rings ==
=== Definition ===

A <em>[[Derivation (differential algebra)|derivation]]</em> <math display="inline"> \partial </math> on a ring <math display="inline"> R </math> is a [[function (mathematics)|function]]
<math>\partial : R \to R\,</math>
such that 
<math display=block>\partial(r_1 + r_2) = \partial r_1 + \partial r_2</math>
and
: <math>\partial(r_1 r_2) = (\partial r_1) r_2 + r_1 (\partial r_2)\quad</math> ([[product rule|Leibniz product rule]]),
for every <math>r_1</math> and <math>r_2</math> in <math>R.</math>

A derivation is [[linear map|linear]] over the integers since these identities imply <math>\partial (0)=\partial (1) = 0</math> and <math>\partial (-r)=-\partial (r).</math>

A ''differential ring'' is a [[commutative ring]] <math>R</math> equipped with one or more derivations that commute pairwise; that is, <math  display=block>\partial_1(\partial_2 (r))=\partial_2(\partial_1 (r))</math> for every pair of derivations and every <math>r\in R.</math>{{sfn|Kolchin|1973|pp=58–59|ps=none}} When there is only one derivation one talks often of an <em>ordinary differential ring</em>; otherwise, one talks of a <em>partial differential ring.</em>

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

A ''Witt algebra'' is a differential ring that contains the field <math>\Q</math> of the rational numbers. Equivalently, this is a differential algebra over <math>\Q,</math> since <math>\Q</math> can be considered as a differential field on which every derivation is the [[zero function]].

The <em>constants</em> of a differential ring are the elements <math>r</math> such that <math>\partial r=0</math> for every derivation <math>\partial.</math> The constants of a differential ring form a [[subring]] and the constants of a differentiable field form a subfield.{{sfn|Kolchin|1973|pp=58–60|ps=none}} This meaning of "constant" generalizes the concept of a [[constant function]], and must not be confused with the common meaning of a [[constant (mathematics)|constant]].

=== Basic formulas ===
In the following [[identity (mathematics)|identities]], <math>\delta</math> is a derivation of a differential ring <math>R.</math>{{sfn|Bronstein|2005|p=76|ps=none}}
* If <math>r\in R</math> and <math>c</math> is a constant in <math>R</math> (that is, <math>\delta c=0</math>), then <math display =block> \delta (c r)= c \delta (r).</math>
* If <math>r\in R</math> and <math>u</math> is a [[Unit (ring theory)|unit]] in <math>R,</math> then  <math display =block> \delta \left( \frac{r}{u} \right)= \frac{\delta (r) u - r \delta (u)}{u^{2}}</math>
* If <math>n</math> is a nonnegative integer and <math>r\in R</math> then <math display =block> \delta (r^{n})= n r^{n-1} \delta (r) </math>
* If <math>u_1, \ldots, u_n</math> are units in <math>R,</math> and <math>e_1, \ldots, e_n</math> are integers, one has the <em>[[logarithmic derivative]] identity:</em> <math display =block> \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}}. </math>

=== Higher-order derivations ===
A <em>derivation operator</em> or <em>higher-order derivation</em>{{citation needed|reason=It is unclear what is the common name in the literature|date=March 2023}} is the [[function composition|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
<math display= block> \delta_1^{e_1} \circ \cdots \circ \delta_n^{e_n},</math>
where <math>\delta_1, \ldots, \delta_n</math> are the derivations under consideration, <math>e_1, \ldots, e_n</math> are nonnegative integers, and the  exponent of a derivation denotes the number of times this derivation is composed in the operator.

The sum <math>o=e_1+ \cdots +e_n</math> is called the ''order'' of derivation. If <math>o=1</math> the derivation operator is one of the original derivations. If <math>o=0</math>, 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 <math>x</math> of a differential ring is the application of a derivation operator to <math>x,</math> that is, with the above notation, <math>\delta_1^{e_1} \circ \cdots \circ \delta_n^{e_n}(x).</math> A <em>proper derivative</em> is a derivative of positive order.{{sfn|Kolchin|1973|pp=58–59|ps=none}}

=== Differential ideals ===
A <em>[[differential ideal]]</em> <math>I</math> of a differential ring <math>R</math> is an [[ideal (ring theory)|ideal]] of the ring <math>R</math> that is [[Closure (mathematics)|closed]] (stable) under the derivations of the ring; that is, <math display="inline"> \partial x\in I,</math> for every derivation <math>\partial</math> and every <math>x\in I.</math> A differential ideal is said to be <em>proper</em> 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 <em>radical</em> of a differential ideal is the same as its [[radical of an ideal|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.{{sfn|Sit|2002|pp=3–4|ps=none}} A prime differential ideal is a differential ideal that is [[prime ideal|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.{{sfn|Kolchin|1973|pp=61–62|ps=none}}
It follows that, given a subset <math>S</math> 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.{{sfn|Kolchin|1973|pp=61–62|ps=none}}{{sfn|Buium|1994|p=21|ps=none}}

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

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

The radical differential ideal generated by <math>S</math> is commonly denoted as <math>\{S\}.</math> 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 <math>K</math> is a formalization of the concept of [[differential equation]] such that the known functions appearing in the equation belong to <math>K,</math> and the indeterminates are symbols for the unknown functions.

So, let <math>K</math> be a differential field, which is typically (but not necessarily) a field of [[rational fraction]]s <math>K(X)=K(x_1,\ldots ,x_n)</math> (fractions of multivariate polynomials), equipped with derivations  <math>\partial_i</math> such that <math>\partial_i x_i=1</math> and <math>\partial_i x_j=0</math> if <math>i\neq j</math> (the usual partial derivatives).

For defining the ring <math display="inline"> K \{ Y \}= K \{ y_1, \ldots, y_n \}</math> of differential polynomials over <math>K</math> with indeterminates in <math>Y=\{y_1,\ldots, y_n\}</math> with derivations <math>\partial_1, \ldots, \partial_n,</math> one introduces an infinity of new indeterminates of the form <math>\Delta y_i,</math> where <math>\Delta</math> is any derivation operator of order higher than {{math|1}}. With this notation, <math>K \{ Y \}</math> 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 <math>n=1,</math> one has
: <math>K\{y\}=K\left[y, \partial y, \partial^2 y, \partial^3 y, \ldots\right].</math>

Even when <math>n=1,</math> a ring of differential polynomials is not [[Noetherian ring|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 divisor]]s exist, and a ring of differential polynomials is a [[unique factorization domain]].

The second fact is that, if the field <math>K</math> contains the field of rational numbers, the rings of differential polynomials over <math>K</math> satisfy the [[ascending chain condition]] on radical differential ideals. This Ritt’s theorem is implied by its generalization, sometimes called the <em>Ritt-Raudenbush basis theorem</em> which asserts that if <math>R</math> is a <em>Ritt Algebra</em> (that, is a differential ring containing the field of rational numbers),{{sfn|Kaplansky|1976|p=12|ps=none}} that satisfies the ascending chain condition on radical differential ideals, then the ring of differential polynomials <math>R\{y\}</math> satisfies the same property (one passes from the univariate to the multivariate case by applying the theorem iteratively).{{sfn|Kaplansky|1976|pp=45,48,56–57|ps=none}}{{sfn|Kolchin|1973|pp=126–129|ps=none}}

This Noetherian property implies that, in a ring of differential polynomials, every radical differential ideal {{mvar|I}} is finitely generated as a radical differential ideal; this means that there exists a finite set {{mvar|S}} of differential polynomials such that {{mvar|I}} is the smallest radical differential idesl containing {{mvar|S}}.{{sfn|Marker|2000|ps=none}} 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 <em>essential prime components</em> of the ideal.{{sfn|Hubert|2002|p=8|ps=none}} <!--

An <em>algebraically independent</em> differential field <math display="inline"> \mathcal{F} \{ Y \} </math> is a differential field with a non-vanishing [[Wronskian | Wronskian determinant]].{{sfn|Bronstein|2005|p=79|ps=none}}

<em>Special</em> and <em>normal</em> polynomials have distinct [[Polynomial greatest common divisor|greatest common divisors]] (gcd) for the polynomial and its derivative.  All irreducible polynomials are special or normal with respect to a derivation; special polynomials may generate a differential ideal while normal polynomials are [[Square-free polynomial | squarefree]].  The definitions are:{{sfn|Bronstein|2005|pp=92–93|ps=none}}
* <em>Normal</em> polynomial <math>p</math>: <math> \ gcd(p,\delta(p))=1</math>.
* <em>Special</em> polynomial <math>q</math>: <math> \ gcd(q,\delta(q))=q</math>.
 -->

== Elimination methods ==
<em>[[Elimination theory|Elimination methods]]</em> 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 <em>[[Wu's method of characteristic set|characteristic set]] methods</em>, differential [[Gröbner basis|Gröbner bases]] methods and [[resultant]] based methods.{{sfn|Kolchin|1973|ps=none}}{{sfn|Li|Yuan|2019|ps=none}}{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|ps=none}}{{sfn|Mansfield|1991|ps=none}}{{sfn|Ferro|2005|ps=none}}{{sfn|Chardin|1991|ps=none}}{{sfn|Wu|2005b|ps=none}}

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 <em>ranking</em> of derivatives is a [[total order]] and an <em>admisible order</em>, defined as:{{sfn|Kolchin|1973|pp=75–76|ps=none}}{{sfn|Gao|Van der Hoeven|Yuan|Zhang|2009|p=1141|ps=none}}{{sfn|Hubert|2002|p=10|ps=none}}
: <math display="inline"> \forall p \in \Theta Y, \ \forall \theta_\mu \in \Theta : \theta_\mu p > p. </math>
: <math display="inline"> \forall p,q \in \Theta Y, \ \forall \theta_\mu \in \Theta : p \ge q \Rightarrow \theta_\mu p \ge \theta_\mu q. </math>

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:{{sfn|Ferro|Gerdt|2003|p=83|ps=none}} 
* <em>Orderly ranking</em>: <math> \forall y_i, y_j \in Y, \ \forall \theta_\mu, \theta_\nu \in \Theta \ : \ \operatorname{ord}(\theta_\mu) \ge \operatorname{ord}(\theta_\nu) \Rightarrow \theta_\mu y_i \ge \theta_\nu y_j</math>
* <em>Elimination ranking</em>: <math>\forall y_i, y_j \in Y, \ \forall \theta_\mu, \theta_\nu \in \Theta \ : \ y_i \ge y_j \Rightarrow \theta_\mu y_i \ge \theta_\nu y_j</math>

In this example, the integer tuple identifies the differential indeterminate and derivative's multi-index, and [[Lexicographic order|lexicographic monomial order]], <math display="inline"> \ge_\text{lex}</math>, determines the derivative's rank.{{sfn|Wu|2005a|p=4|ps=none}}
: <math>\eta(\delta_1^{e_1} \circ \cdots \circ \delta_n^{e_n}(y_j))= (j, e_1, \ldots, e_n) </math>. 
: <math> \eta(\theta_\mu y_j) \ge_\text{lex} \eta(\theta_\nu y_k) \Rightarrow \theta_\mu y_j \ge \theta_\nu y_k. </math>

=== Leading derivative, initial and separant ===
This is the standard polynomial form: <math> p = a_d \cdot u_p^d+ a_{d-1} \cdot u_p^{d-1} + \cdots +a_1 \cdot u_p+ a_0 </math>.{{sfn|Kolchin|1973|pp=75–76|ps=none}}{{sfn|Wu|2005a|p=4|ps=none}} 
* <em>Leader</em> or <em>leading derivative</em> is the polynomial's highest ranked derivative: <math>u_p</math>. 
* [[Coefficient]]s <math>a_d, \ldots, a_0</math> do not contain the leading derivative <math display="inline">u_p</math>.
* <em>[[Degree of a polynomial|Degree]]</em> of polynomial is the leading derivative's greatest exponent: <math>\deg_{u_p}(p) = d</math>. 
* <em>Initial</em> is the coefficient: <math> I_p=a_d</math>. 
* <em>Rank</em> is the leading derivative raised to the polynomial's degree: <math>u_p^d</math>. 
* <em>[[Differentially closed field|Separant]]</em> is the derivative: <math> S_p= \frac{\partial p}{\partial u_p}</math>.

Separant set is <math>S_A= \{ S_p \mid p \in A \} </math>, initial set is <math>I_A= \{ I_p \mid p \in A \} </math> and combined set is <math display="inline">H_A= S_A \cup I_A </math>.{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=159|ps=none}}

=== Reduction ===
<em>Partially reduced</em> (<em>partial normal form</em>) polynomial <math display="inline">q</math> with respect to polynomial <math display="inline">p</math> indicates these polynomials are non-ground field elements, <math display="inline"> p,q \in \mathcal{K} \{ Y \} \setminus \mathcal{K}</math>, and <math>q</math> contains no proper derivative of <math> u_p</math>.{{sfn|Kolchin|1973|p=75|ps=none}}{{sfn|Ferro|Gerdt|2003|p=84|ps=none}}{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=159|ps=none}}

Partially reduced polynomial <math display="inline">q</math> with respect to polynomial <math display="inline">p</math> becomes 
<em>reduced</em> (<em>normal form</em>) polynomial <math display="inline">q</math> with respect to <math display="inline">p</math> if the degree of <math display="inline">u_p</math> in <math display="inline">q</math> is less than the degree of <math display="inline">u_{p}</math> in <math display="inline">p</math>.{{sfn|Kolchin|1973|p=75|ps=none}}{{sfn|Ferro|Gerdt|2003|p=84|ps=none}}{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=159|ps=none}}

An <em>autoreduced</em> polynomial set has every polynomial reduced with respect to every other polynomial of the set.  Every autoreduced set is finite.  An autoreduced set is <em>[[Triangular decomposition|triangular]]</em> meaning each polynomial element has a distinct leading derivative.{{sfn|Sit|2002|p=6|ps=none}}{{sfn|Kolchin|1973|p=75|ps=none}}

<em>Ritt's reduction algorithm</em> identifies integers <math display="inline">i_{A_{k}}, s_{A_{k}}</math> and transforms a differential polynomial <math display="inline">f</math> using [[Polynomial greatest common divisor|pseudodivision]] to a lower or equally ranked remainder polynomial <math display="inline"> f_{red}</math> that is reduced with respect to the autoreduced polynomial set <math display="inline"> A</math>.  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:{{sfn|Kolchin|1973|p=75|ps=none}}
: <math> 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}. </math>

=== Ranking polynomial sets ===
Set <math display="inline">A</math> is a <em>differential chain</em> if the rank of the leading derivatives is <math display="inline">u_{A_{1}} < \dots < u_{A_{m}} </math> and <math display="inline">\forall i, \ A_{i}</math> is reduced with respect to <math display="inline">A_{i+1}</math>{{sfn|Li|Yuan|2019|p=294|ps=none}}

Autoreduced sets <math display="inline">A</math> and <math display="inline">B</math> each contain ranked polynomial elements.  This procedure ranks two autoreduced sets by comparing pairs of identically indexed 
polynomials from both autoreduced sets.{{sfn|Kolchin|1973|p=81|ps=none}} 
* <math>A_1 < \cdots < A_m \in A </math> and <math>B_1 < \cdots < B_n \in B </math> and <math> i,j,k \in \mathbb{N}</math>.
* <math> \text{rank } A < \text{rank } B </math> if there is a <math> k \le \operatorname{minimum}(m,n) </math> such that <math> A_i = B_i</math> for <math display="inline"> 1 \le i < k </math> and <math> A_k < B_k </math>.
* <math> \operatorname{rank} A < \operatorname{rank} B </math> if <math> n < m </math> and <math>A_i = B_i</math> for <math>1 \le i \le n </math>.
* <math> \operatorname{rank} A = \operatorname{rank} B </math> if <math> n = m </math> and <math>A_i = B_i</math> for <math>1 \le i \le n </math>.

=== Polynomial sets ===
A <em>characteristic set</em> <math display="inline">C</math> is the [[Arg max|lowest ranked]] autoreduced subset among all the ideal's autoreduced subsets whose subset polynomial separants are non-members of the ideal <math display="inline">\mathcal{I}</math>.{{sfn|Kolchin|1973|p=82|ps=none}}

The <em>delta polynomial</em> applies to polynomial pair <math display="inline">p,q</math> whose leaders share a common derivative, <math display="inline">\theta_{\alpha} u_{p}= \theta_{\beta} u_{q}</math>.  The least common derivative operator for the polynomial pair's leading derivatives is <math display="inline">\theta_{pq}</math>, and the delta polynomial is:{{sfn|Kolchin|1973|p=136|ps=none}}{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=160|ps=none}} 
: <math>\operatorname{\Delta - poly}(p,q)= S_{q} \cdot \frac{\theta_{pq} p}{\theta_{p}} - S_{p} \cdot \frac{\theta_{pq} q}{\theta_{q}} </math>

A <em>coherent set</em> is a polynomial set that reduces its delta polynomial pairs to zero.{{sfn|Kolchin|1973|p=136|ps=none}}{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=160|ps=none}}

=== Regular system and regular ideal ===
A <em>regular system</em> <math display="inline">\Omega</math> contains a autoreduced and coherent set of differential equations <math display="inline">A</math> and a inequation set <math display="inline">H_{\Omega} \supseteq H_A</math> with set <math display="inline">H_\Omega </math> reduced with respect to the equation set.{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=160|ps=none}}

Regular differential ideal <math display="inline">\mathcal{I}_\text{dif} </math> and regular algebraic ideal <math display="inline">\mathcal{I}_\text{alg} </math> are [[Ideal quotient|saturation ideals]] that arise from a regular system.{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=160|ps=none}}  <em>Lazard's lemma</em> states that the regular differential and regular algebraic ideals are radical ideals.{{sfn|Morrison|1999|ps=none}}
* <em>Regular differential ideal</em>: <math display="inline">\mathcal{I}_\text{dif}=[A]:H_\Omega^\infty.</math>
* <em>Regular algebraic ideal</em>: <math display="inline">\mathcal{I}_\text{alg}=(A):H_\Omega^\infty.</math>

=== Rosenfeld–Gröbner algorithm ===
The <em>Rosenfeld–Gröbner algorithm</em> 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 [[Primary decomposition|minimal]].{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=158|ps=none}}

The <em>membership problem</em> is to determine if a differential polynomial <math display="inline">p</math> is a member of an ideal generated from a set of differential polynomials <math display="inline">S</math>. 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.{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=164|ps=none}}

The Rosenfeld–Gröbner algorithm facilitates creating [[Taylor series]] expansions of solutions to the differential equations.{{sfn|Boulier|Lazard|Ollivier|Petitot|2009b|ps=none}}

== Examples ==
=== Differential fields ===
Example 1: <math display="inline">(\operatorname{Mer}(\operatorname{f}(y), \partial_{y} ))</math> is the differential [[meromorphic function]] field with a single <em>standard derivation</em>.

Example 2: <math display="inline">(\mathbb{C} \{ y \}, p(y)\cdot \partial_{y} ) </math> is a differential field with a [[Differential operator|linear differential operator]] as the derivation, for any polynomial <math>p(y) </math>.

=== Derivation ===
Define <math display="inline">E^{a}(p(y))=p(y+a)</math> as <em>[[shift operator]]</em> <math display="inline">E^{a}</math> for polynomial <math display="inline">p(y)</math>.

A shift-invariant operator <math display="inline">T</math> commutes with the shift operator: <math display="inline">E^{a} \circ T=T \circ E^{a}</math>.

The <em>[[Pincherle derivative]]</em>, a derivation of shift-invariant operator <math display="inline">T</math>, is <math display="inline">T^{\prime} = T \circ y - y \circ T </math>.{{sfn|Rota|Kahaner|Odlyzko|1973|p=694|ps=none}}

=== Constants ===
Ring of integers is <math>(\mathbb{Z}. \delta)</math>, and every integer is a constant. 
* The derivation of 1 is zero. <math display="inline"> \delta(1)=\delta(1 \cdot 1)=\delta(1) \cdot 1 + 1 \cdot \delta(1) = 2 \cdot \delta(1) \Rightarrow \delta(1)=0</math>.
* Also, <math> \delta(m+1)=\delta(m)+\delta(1)=\delta(m) \Rightarrow \delta(m+1)=\delta(m) </math>. 
* By induction, <math> \delta(1)=0 \ \wedge \ \delta(m+1)= \delta(m) \Rightarrow \forall \ m \in \mathbb{Z}, \ \delta(m)=0 </math>.

Field of rational numbers is <math>(\mathbb{Q}. \delta)</math>, and every rational number is a constant. 
* Every rational number is a quotient of integers.
*: <math> \forall r \in \mathbb{Q}, \ \exists \ a \in \mathbb{Z}, \ b \in \mathbb{Z}/ \{ 0 \}, \ r=\frac{a}{b} </math>
* Apply the derivation formula for quotients recognizing that derivations of integers are zero:
*: <math> \delta (r)= \delta \left ( \frac{a}{b} \right ) = \frac{\delta(a) \cdot b - a \cdot \delta(b)}{b^{2}}=0 </math>.

=== Differential subring ===
Constants form the <em>subring of constants</em> <math display="inline">(\mathbb{C}, \partial_{y}) \subset (\mathbb{C} \{ y \}, \partial_{y}) </math>.{{sfn|Kolchin|1973|p=60|ps=none}}

=== Differential ideal ===
Element <math display="inline">\exp(y)</math> simply generates differential ideal <math display="inline"> [\exp(y)] </math> in the differential ring <math display="inline">(\mathbb{C} \{ y, \exp(y) \}, \partial_{y}) 
</math>.{{sfn|Sit|2002|p=4|ps=none}}

=== Algebra over a differential ring ===
Any ring with identity is a <math display="inline">\operatorname{\mathcal{Z}-}</math>algebra.{{sfn|Dummit|Foote|2004|p=343|ps=none}}  Thus a differential ring is a <math display="inline">\operatorname{\mathcal{Z}-}</math>algebra.

If ring <math display="inline">\mathcal{R}</math> is a subring of the center of unital ring <math display="inline">\mathcal{M}</math>, then <math display="inline">\mathcal{M}</math> is an <math display="inline">\operatorname{\mathcal{R}-}</math>algebra.{{sfn|Dummit|Foote|2004|p=343|ps=none}}  Thus, a differential ring is an algebra over its differential subring.  This is the <em>natural structure</em> of an algebra over its subring.{{sfn|Kolchin|1973|p=75|ps=none}}

=== Special and normal polynomials ===
Ring <math display="inline">(\mathbb{Q} \{ y, z \}, \partial_y) </math> has irreducible polynomials, <math display="inline">p</math> (normal, squarefree) and <math display="inline">q</math> (special, ideal generator). 
: <math display="inline"> \partial_y(y)=1, \ \partial_y(z)=1+z^2, \ z=\tan(y)</math> 
: <math display="inline">p(y)=1+y^2, \ \partial_y(p)=2 \cdot y,\ \gcd(p, \partial_y(p))=1</math>
: <math display="inline">q(z)=1+z^2, \ \partial_y(q)=2 \cdot z \cdot (1+z^2),\ \gcd(q, \partial_{y}(q))=q</math>

=== Polynomials ===
==== Ranking ====
Ring <math display="inline">(\mathbb{Q} \{ y_{1}, y_{2} \}, \delta)</math> has derivatives <math display="inline">\delta(y_{1})=y_{1}^{\prime}</math> and <math display="inline">\delta(y_{2})=y_{2}^{\prime}</math> 
* Map each derivative to an integer tuple: <math display="inline">\eta( \delta^{(i_{2})}(y_{i_{1}}) )=(i_{1}, i_{2})</math>.
* Rank derivatives and integer tuples: <math display="inline"> 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) </math>.

==== Leading derivative and initial ====
The <span style="color:red">leading derivatives</span>, and <span style="color:blue">initials</span> are:
: <math display="inline"> 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} </math> 
: <math display="inline"> 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} </math> 
: <math display="inline"> 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} </math>

==== Separants ====
: <math display="inline"> S_{p}= 2 \cdot (y_{1}+ y_{1}^{\prime}) \cdot y_{2}^{\prime \prime} + 3 \cdot y_{1}^{2}</math>. 
: <math display="inline"> S_{q}= y_{1}+ 3 \cdot y_{1}^{\prime}</math>
: <math display="inline"> S_{r}= 2 \cdot (y_{1}+3) \cdot y_{1}^{\prime \prime} + y_{1}^{2}</math>

==== Autoreduced sets ====
* Autoreduced sets are <math display="inline">\{ p, r \}</math> and <math display="inline"> \{ q, r \}</math>.  Each set is triangular with a distinct polynomial leading derivative.
* The non-autoreduced set <math display="inline"> \{ p, q \} </math> contains only partially reduced <math display="inline">p</math> with respect to <math display="inline">q</math>; 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.{{sfn|Bronstein|2005|pp=41,51,53,102,299,309|ps=none}}

=== 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|differential-algebraic]] system of equations.{{sfn|Hubert|2002|pp=41–47|ps=none}}

In a study of non-linear dynamical systems with [[Chaos theory|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 function|Lyapunov functions]].{{sfn|Harrington|VanGorder|2017|ps=none}}  Researchers have applied differential elimination to understanding [[Cell biology|cellular biology]], [[Physiologically based pharmacokinetic modelling|compartmental biochemical models]], [[parameter]] estimation and [[Steady state (chemistry)|quasi-steady state]] approximation (QSSA) for biochemical reactions.{{sfn|Boulier|2007|ps=none}}{{sfn|Boulier|Lemaire| 2009a|ps=none}}  Using differential Gröbner bases, researchers have investigated non-classical [[Symmetry in mathematics|symmetry]] properties of [[Nonlinear system|non-linear differential equations]].{{sfn|Clarkson|Mansfield|1994|ps=none}}  Other applications include control theory, model theory, and algebraic geometry.{{sfn|Diop|1992|ps=none}}{{sfn|Marker|2000|ps=none}}{{sfn|Buium|1994|ps=none}}  Differential algebra also applies to differential-difference equations.{{sfn|Gao|Van der Hoeven|Yuan|Zhang|2009|ps=none}}<!--

== Linear differential algebra ==
{{hatnote|Most of the content of this section consists of trivialities that are not useful for understanding the remainder of the article, except the sentence on differentially closed fields, which deserves to have a specific section.}}
=== Subrings ===
The <math display="inline">\operatorname{\Delta_{R} - \mathcal{R}}</math> is a <em>differential subring</em> of <math display="inline"> \operatorname{\Delta_{S} - \mathcal{S}} </math> if <math display="inline"> \mathcal{R}</math> is a [[subring]] of <math display="inline"> \mathcal{S}</math>, and the derivation set <math display="inline"> \operatorname{\Delta_{R}} </math> is the derivation set <math display="inline"> \operatorname{\Delta_{S}} </math> [[Restriction (mathematics) |restricted]] to <math display="inline"> \mathcal{R}</math>.  An equivalent statement is <math display="inline"> \operatorname{\Delta_{S} - \mathcal{S}} </math> is the differential [[overring]] of <math display="inline"> \operatorname{\Delta_{R} - \mathcal{R}} </math>.{{sfn|Kolchin|1973|pp=58–59|ps=none}}

The <em>[[Intersection (set theory) | intersection]]</em> of any family of differential subrings is a differential subring.  The intersection of any set of differential subrings containing a <em>common set</em> is a differential subring, and the <em>smallest differential subring</em> containing a common set is the intersection of all subrings containing the common set.{{sfn|Kolchin|1973|pp=58–59|ps=none}}

Set <math display="inline"> \Theta A </math> generates differential ring <math display="inline">\mathcal{R} \{ A \} </math> over <math display="inline">\mathcal{R}</math>.  This is the smallest differential subring containing differential subring <math display="inline">\mathcal{R} </math> and set <math display="inline"> \Theta A </math>.  A <em>finitely generated</em> differential subring arises from a finite set, and a <em>simply generated</em> differential subring arises from a single element.  <em>Adjoining</em> or adding an element to the generator set extends the differential ring.  Using the square bracket notation for ring extension, <math display="inline">\mathcal{R} \{ A \}=\mathcal{R} [ \Theta A ] </math>.{{sfn|Kolchin|1973|pp=58–60|ps=none}}

Set <math display="inline"> \Theta A </math> generates differential field <math display="inline">\mathcal{F} \langle A \rangle </math> over field <math display="inline">\mathcal{F}</math>.  Using the parentheses notation for a field extension, <math display="inline">\mathcal{F} \langle A \rangle =\mathcal{F} ( \Theta A ) </math>.{{sfn|Kolchin|1973|p=60|ps=none}}

A field <math display="inline">K</math> is a <em>[[Differentially closed field|closed differential field]]</em> if each instance when a differential equation set's solution, <math display="inline">f_{i} \in K \{ y_{1}, \ldots y_{m} \}</math> for <math display="inline">i \in \{ 1, \ldots, m \}</math>, occurs in field <math display="inline">L</math> extended over <math display="inline">K</math>, the  solution occurs in the field <math display="inline">K</math>.{{sfn|Marker|2000|ps=none}}{{rp|54}}  Any differential field may extend to a closed differential field.{{sfn|Marker|2000|p=54|ps=none}}  [[Differential Galois theory]] studies differential field extensions and the associated Galois group.{{sfn|Crespo|Hajto|2011|p=141|ps=none}}

=== Ring homomorphism ===
A <em>differential [[Homomorphism |ring homomorphism]]</em> is a map, <math display="inline"> \operatorname{f}: \mathcal{R} \to \mathcal{S} </math> of differential rings that share the same derivation set, <math display="inline"> \Delta_{R}=\Delta_{S} </math>, and the ring homomorphism commutes with derivation, <math display="inline"> \forall r \in \mathcal{R}, \ \forall \delta \in \Delta \ : \ \delta (\operatorname{f}(r))= \operatorname{f}(\delta(r)) </math>.{{sfn|Kolchin|1973|p=61|ps=none}}
* The [[Kernel (algebra) | kernel]] is a differential ideal of <math display="inline"> \mathcal{R}</math>, and the [[Image (mathematics) |image]] is a differential subring.{{sfn|Kolchin|1973|p=61|ps=none}} 
* The ring <math display="inline"> \mathcal{S}</math> is an [[Ring extension |extension]] of <math display="inline"> \mathcal{R}</math>, and <math display="inline"> \mathcal{R}</math> is a subring of <math display="inline"> \mathcal{S}</math> if the ring homomorphism is an [[Subset | inclusion]].{{sfn|Buium|1994|p=21|ps=none}} 
* For differential ring <math display="inline">\mathcal{R} </math> and differential ideal <math display="inline">\mathcal{I} </math>, the <em>[[Canonical map | canonical homomorphism]]</em> maps the ring to the differential [[Quotient ring | residue ring]]: <math display="inline"> \operatorname{f}: \mathcal{R} \to \mathcal{R} / \mathcal{I} </math>.

=== Modules ===
A <em>differential <math display="inline"> \operatorname{\mathcal{R} - module}</math></em> or [[Module (mathematics)|module]] over differential ring <math display="inline"> \operatorname{\Delta - \mathcal{R}} </math> has module <math display="inline"> \mathcal{M} </math> whose elements follow these sum and product derivation rules: <math display="inline"> \delta \in \Delta, \ r \in \mathcal{R}, \ u,v \in \mathcal{M} </math>:{{sfn|Kolchin|1973|p=66|ps=none}} 
: <math display="inline"> \delta(u+v)= \delta (u) + \delta (v) </math> 
: <math display="inline"> \delta(r \cdot u)= \delta (r) \cdot u + r \cdot \delta (u) </math>

A <em>differential [[vector space]]</em> is a differential module over a differential field.

A <em>differential <math display="inline"> \operatorname{\mathcal{R}-algebra}</math></em> or [[Associative algebra | differential algebra]] over the <math display="inline"> \mathcal{R} </math> is the ring <math display="inline"> \mathcal{M} </math>, the <math display="inline"> \operatorname{\mathcal{R}-algebra}</math>, and a derivation set <math display="inline">\Delta</math> that makes <math display="inline"> \mathcal{M} </math> a differential ring and that follows this derivation product rule:{{sfn|Kolchin|1973|p=69|ps=none}}{{sfn|Dummit|Foote|2004|p=342|ps=none}}
: <math> \forall \delta \in \Delta, \ \forall r \in \mathcal{R}, \ \forall u \in \mathcal{M} \ : \ \delta(r \cdot u)= \delta (r) \cdot u + r \cdot \delta (u) </math>. -->

== Algebras with derivations ==
=== Differential graded vector space ===
A <em><math display="inline">\operatorname{\mathbb{Z}-graded}</math> [[Graded vector space|vector space]]</em> <math display="inline">V_{\bullet} </math> is a collection of vector spaces <math display="inline">V_{m}</math> with integer <em>degree</em> <math display="inline">|v|=m</math> for <math display="inline"> v\in V_{m}</math>.  A [[direct sum]] can represent this graded vector space:{{sfn|Keller|2019|p=48|ps=none}}
: <math>V_{\bullet} = \bigoplus_{m \in \mathbb{Z}} V_{m}</math>

A <em>differential graded vector space</em> or <em>[[chain complex]]</em>, is a graded vector space <math display="inline">V_{\bullet}</math> with a <em>differential map</em> or <em>boundary map</em> <math display="inline">d_{m}: V_{m} \to V_{m-1}</math> with <math> d_{m} \circ d_{m+1} = 0 </math> .{{sfn|Keller|2019|pp=50–51|ps=none}}

A <em>[[Chain complex|cochain complex]]</em> is a graded vector space <math display="inline">V^{\bullet}</math> with a <em>differential map</em> or <em>coboundary map</em> 
<math display="inline">d_{m}: V_{m} \to V_{m+1}</math> with <math> d_{m+1} \circ d_{m} = 0 </math>.{{sfn|Keller|2019|pp=50–51|ps=none}}

=== Differential graded algebra ===
A <em>differential graded algebra</em> is a graded algebra <math display="inline">A</math> with a linear derivation <math display="inline">d: A \to A </math> with <math>d \circ d=0 </math> that follows the graded Leibniz product rule.{{sfn|Keller|2019|pp=58–59|ps=none}}
* Graded Leibniz product rule: <math>\forall a,b \in A, \ d(a \cdot b)=d(a) \cdot b + (-1)^{|a|} \cdot a \cdot d(b)</math> with <math>|a|</math> the degree of vector <math>a</math>.

=== Lie algebra ===
A <em>[[Lie algebra]]</em> is a finite-dimensional real or complex vector space <math display="inline">\mathcal{g}</math> with a [[Bilinear form|bilinear]] bracket operator <math display="inline">[,]:\mathcal{g} \times \mathcal{g} \to \mathcal{g} </math> with [[Bilinear form|Skew symmetry]] and the [[Jacobi identity]] property.{{sfn|Hall|2015|p=49|ps=none}}
* Skew symmetry: <math> [X,Y]= -[Y,X]</math>
* Jacobi identity property: <math> [X,[Y,Z]]+[Y,[Z,X]] + [Z,[X,Y]]=0 </math>
for all <math> X, Y, Z \in \mathcal{g}</math>.

The <em>adjoint</em> operator, <math display="inline">\operatorname{ad_{X}}(Y)=[Y,X]</math> is a <em>[[Commutator|derivation of the bracket]]</em> 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 <em>inner derivation</em> determined by <math display="inline">X</math>.{{sfn|Hall|2015|p=51|ps=none}}{{sfn|Jacobson|1979|p=9|ps=none}}  
: <math> \operatorname{ad}_{X}([Y,Z]) = [\operatorname{ad}_{X}(Y),Z] + [Y,\operatorname{ad}_{X}(Z)] </math>

The <em>[[universal enveloping algebra]]</em> <math display="inline">U(\mathcal{g})</math> of Lie algebra <math display="inline">\mathcal{g}</math> is a maximal associative algebra with identity, generated by Lie algebra elements <math display="inline">\mathcal{g}</math> 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.{{sfn|Hall|2015|p=247|ps=none}} 
* Product in <math>U(\mathcal{g})</math> : <math>X \cdot Y - Y \cdot X = [X,Y]</math>
* Leibniz product rule: <math>\operatorname{ad}_{X}( Y \cdot Z)=\operatorname{ad}_{X}(Y) \cdot Z + Y \cdot \operatorname{ad}_{X}(Z)</math>
for all <math> X,Y,Z \in U(\mathcal{g}) </math>.

=== Weyl algebra ===
The [[Weyl algebra]] is an algebra <math display="inline">A_{n}(K)</math>  over a ring <math display="inline">K [p_{1}, q_{1}, \dots, p_{n}, q_{n}]</math> with a specific noncommutative product: {{sfn|Lam|1991|pp=7–8|ps=none}} 
: <math> p_{i} \cdot q_{i} - q_{i} \cdot p_{i}=1, \ : \ i \in \{1, \dots, n \} </math>.
All other indeterminate products are commutative for <math display="inline">i,j \in \{1, \dots, n \}</math>: 
: <math> p_{i} \cdot q_{j} - q_{j} \cdot p_{i}=0 \text{  if  } i \ne j, \ p_{i} \cdot p_{j} - p_{j} \cdot p_{i}=0, \ q_{i} \cdot q_{j} - q_{j} \cdot q_{i}=0 </math>.
A Weyl algebra can represent the derivations for a commutative ring's polynomials <math display="inline">f \in K[y_{1}, \ldots, y_{n}]</math>.  The Weyl algebra's elements are [[endomorphism]]s, the elements <math display="inline">p_{1}, \ldots, p_{n}</math> function as standard derivations, and map compositions generate [[Differential operator|linear differential operators]].  [[D-module]] is a related approach for understanding differential operators. The endomorphisms are:{{sfn|Lam|1991|pp=7–8|ps=none}}
: <math> 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 \ne k, \ p_{j}(c)= 0 \text{  with  } c \in K </math>

=== Pseudodifferential operator ring ===
The associative, possibly noncommutative ring <math display="inline">A</math> has derivation <math display="inline">d: A \to A </math>.{{sfn|Parshin|1999|p=268|ps=none}}

The <em>[[pseudo-differential operator]] ring</em> <math display="inline">A((\partial^{-1}))</math> is a left <math display="inline">\operatorname{A-module}</math> 
containing ring elements <math display="inline">L</math>:{{sfn|Parshin|1999|p=268|ps=none}}{{sfn|Dummit|Foote|2004|p=337|ps=none}}{{sfn|Taylor|1991|ps=none}}  
: <math> a_i \in A, \ i,i_{\min} \in \mathbb{N}, \ |i_{\min}| > 0 \ : \ L= \sum_{i \ge i_{\min}}^n a_i \cdot \partial^i</math>

The derivative operator is <math display="inline"> d(a) = \partial \circ a - a \circ \partial </math>.{{sfn|Parshin|1999|p=268|ps=none}}

The [[binomial coefficient]] is <math>\Bigl( {i \atop k} \Bigr)</math>.

Pseudo-differential operator multiplication is:{{sfn|Parshin|1999|p=268|ps=none}}
: <math>\sum_{i \ge i_{\min}}^n a_i \cdot \partial^i \cdot \sum_{j\ge j_{\min}}^m b_{i} \cdot \partial^j = \sum_{i,j;k \ge 0} \Bigl( {i \atop k} \Bigr) \cdot a_i \cdot d^k(b_j) \cdot \partial^{i+j-k}</math>

== Open problems ==
The <em>Ritt problem</em> asks is there an algorithm that determines if one prime differential ideal contains a second prime differential ideal when characteristic sets identify both ideals.{{sfn|Golubitsky|Kondratieva|Ovchinnikov|2009|ps=none}}

The <em>Kolchin catenary conjecture</em> states given a <math display="inline">d>0</math> dimensional irreducible differential algebraic variety <math display="inline"> V</math> and an arbitrary point <math display="inline"> p \in V</math>, a long gap chain of irreducible differential algebraic subvarieties occurs from <math display="inline"> p </math> to V.{{sfn|Freitag|Sánchez|Simmons|2016|ps=none}}

The <em>[[Jacobi bound problem|Jacobi bound conjecture]]</em> 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}}

== See also ==
* {{annotated link|Arithmetic derivative}}
* {{annotated link|Difference algebra}}
* {{annotated link|Differential algebraic geometry}}
* {{annotated link|Differential calculus over commutative algebras}}
* {{annotated link|Differential Galois theory}}
* {{annotated link|Differentially closed field}}
* {{annotated link|Differential graded algebra}}
* {{annotated link|D-module}}
* {{annotated link|Hardy field}}
* {{annotated link|Kähler differential}}
* {{annotated link|Liouville's theorem (differential algebra)}}
* {{annotated link|Picard–Vessiot theory}}
* [[Kolchin's problems]]

== Citations ==
{{reflist|colwidth=30em}}

== References ==
{{refbegin|35em}}
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|doi=10.1007/3-7643-7429-2_19 |chapter-url=https://link.springer.com/chapter/10.1007/3-7643-7429-2_19 |language=en}}
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{{refend}}

== External links ==
* [http://www.math.uic.edu/~marker/ David Marker's home page] has several online surveys discussing differential fields.

[[Category:Differential algebra| ]]