Differential algebra

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 Manifolds Of Functions Defined By Systems Of Algebraic Differential Equations and 2 books, Differential Equations From The Algebraic Standpoint and Differential Algebra.{{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 Differential Algebra And Algebraic Groups.{{sfn|Kolchin|1973|ps=none}}

A derivation $\backslash partial$ on a ring $R$ is a function

$\backslash partial\; :\; R\; \backslash to\; R\backslash ,$

such that

$$\backslash partial(r\_1\; +\; r\_2)\; =\; \backslash partial\; r\_1\; +\; \backslash partial\; r\_2$$

and

: $\backslash partial(r\_1\; r\_2)\; =\; (\backslash partial\; r\_1)\; r\_2\; +\; r\_1\; (\backslash partial\; r\_2)\backslash quad$ (Leibniz product rule),

for every $r\_1$ and $r\_2$ in $R.$

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

A differential ring is a commutative ring $R$ equipped with one or more derivations that commute pairwise; that is, $$\backslash partial\_1(\backslash partial\_2\; (r))=\backslash partial\_2(\backslash partial\_1\; (r))$$ for every pair of derivations and every $r\backslash in\; R.${{sfn|Kolchin|1973|pp=58–59|ps=none}} 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 $A$ over a differential field $K$ is a differential ring that contains $K$ as a subring such that the restriction to $K$ of the derivations of $A$ equal the derivations of $K.$ (A more general definition is given below, which covers the case where $K$ is not a field, and is essentially equivalent when $K$ is a field.)

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

The constants of a differential ring are the elements $r$ such that $\backslash partial\; r=0$ for every derivation $\backslash partial.$ 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.

In the following identities, $\backslash delta$ is a derivation of a differential ring $R.${{sfn|Bronstein|2005|p=76|ps=none}}

• If $r\backslash in\; R$ and $c$ is a constant in $R$ (that is, $\backslash delta\; c=0$), then $$\backslash delta\; (c\; r)=\; c\; \backslash delta\; (r).$$
• If $r\backslash in\; R$ and $u$ is a unit in $R,$ then $$\backslash delta\; \backslash left(\; \backslash frac\{r\}\{u\}\; \backslash right)=\; \backslash frac\{\backslash delta\; (r)\; u\; -\; r\; \backslash delta\; (u)\}\{u^\{2\}\}$$
• If $n$ is a nonnegative integer and $r\backslash in\; R$ then $$\backslash delta\; (r^\{n\})=\; n\; r^\{n-1\}\; \backslash delta\; (r)$$
• If $u\_1,\; \backslash ldots,\; u\_n$ are units in $R,$ and $e\_1,\; \backslash ldots,\; e\_n$ are integers, one has the logarithmic derivative identity: $$\backslash frac\{\backslash delta\; (u\_\{1\}^\{e\_\{1\}\}\; \backslash ldots\; u\_\{n\}^\{e\_\{n\}\})\}\{u\_\{1\}^\{e\_\{1\}\}\; \backslash ldots\; u\_\{n\}^\{e\_\{n\}\}\}\; =\; e\_\{1\}\; \backslash frac\{\backslash delta(\; u\_\{1\}\; )\; \}\{u\_\{1\}\}\; +\; \backslash dots\; +\; e\_\{n\}\; \backslash frac\{\backslash delta(\; u\_\{n\}\; )\; \}\{u\_\{n\}\}.$$

A derivation operator or higher-order derivation{{citation needed|reason=It is unclear what is the common name in the literature|date=March 2023}} 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

$$\backslash delta\_1^\{e\_1\}\; \backslash circ\; \backslash cdots\; \backslash circ\; \backslash delta\_n^\{e\_n\},$$

where $\backslash delta\_1,\; \backslash ldots,\; \backslash delta\_n$ are the derivations under consideration, $e\_1,\; \backslash ldots,\; e\_n$ are nonnegative integers, and the exponent of a derivation denotes the number of times this derivation is composed in the operator.

The sum $o=e\_1+\; \backslash cdots\; +e\_n$ is called the order of derivation. If $o=1$ the derivation operator is one of the original derivations. If $o=0$, one has the identity function, which is generally considered as the unique derivation operator of order zero. With these conventions, the derivation operators form a free commutative monoid on the set of derivations under consideration.

A derivative of an element $x$ of a differential ring is the application of a derivation operator to $x,$ that is, with the above notation, $\backslash delta\_1^\{e\_1\}\; \backslash circ\; \backslash cdots\; \backslash circ\; \backslash delta\_n^\{e\_n\}(x).$ A proper derivative is a derivative of positive order.{{sfn|Kolchin|1973|pp=58–59|ps=none}}

A differential ideal $I$ of a differential ring $R$ is an ideal of the ring $R$ that is closed (stable) under the derivations of the ring; that is, $\backslash partial\; x\backslash in\; I,$ for every derivation $\backslash partial$ and every $x\backslash in\; I.$ 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.{{sfn|Sit|2002|pp=3–4|ps=none}} 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.{{sfn|Kolchin|1973|pp=61–62|ps=none}}

It follows that, given a subset $S$ of a differential ring, there are three ideals generated by it, which are the intersections of, respectively, all algebraic ideals, all differential ideals, and all radical differential ideals that contain it.{{sfn|Kolchin|1973|pp=61–62|ps=none}}{{sfn|Buium|1994|p=21|ps=none}}

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

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

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

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

So, let $K$ be a differential field, which is typically (but not necessarily) a field of rational fractions $K(X)=K(x\_1,\backslash ldots\; ,x\_n)$ (fractions of multivariate polynomials), equipped with derivations $\backslash partial\_i$ such that $\backslash partial\_i\; x\_i=1$ and $\backslash partial\_i\; x\_j=0$ if $i\backslash neq\; j$ (the usual partial derivatives).

For defining the ring $K\; \backslash \{\; Y\; \backslash \}=\; K\; \backslash \{\; y\_1,\; \backslash ldots,\; y\_n\; \backslash \}$ of differential polynomials over $K$ with indeterminates in $Y=\backslash \{y\_1,\backslash ldots,\; y\_n\backslash \}$ with derivations $\backslash partial\_1,\; \backslash ldots,\; \backslash partial\_n,$ one introduces an infinity of new indeterminates of the form $\backslash Delta\; y\_i,$ where $\backslash Delta$ is any derivation operator of order higher than {{math|1}}. With this notation, $K\; \backslash \{\; Y\; \backslash \}$ 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 $n=1,$ one has

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

Even when $n=1,$ 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 $K$ contains the field of rational numbers, the rings of differential polynomials over $K$ satisfy the ascending chain condition on radical differential ideals. This Ritt’s theorem is implied by its generalization, sometimes called the Ritt-Raudenbush basis theorem which asserts that if $R$ is a Ritt Algebra (that, is a differential ring containing the field of rational numbers),{{sfn|Kaplansky|1976|p=12|ps=none}} that satisfies the ascending chain condition on radical differential ideals, then the ring of differential polynomials $R\backslash \{y\backslash \}$ 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 ideal 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 essential prime components of the ideal.{{sfn|Hubert|2002|p=8|ps=none}}

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.{{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.

The ranking of derivatives is a total order and an admisible order, 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}}

: $\backslash forall\; p\; \backslash in\; \backslash Theta\; Y,\; \backslash \; \backslash forall\; \backslash theta\_\backslash mu\; \backslash in\; \backslash Theta\; :\; \backslash theta\_\backslash mu\; p\; >\; p.$

: $\backslash forall\; p,q\; \backslash in\; \backslash Theta\; Y,\; \backslash \; \backslash forall\; \backslash theta\_\backslash mu\; \backslash in\; \backslash Theta\; :\; p\; \backslash ge\; q\; \backslash Rightarrow\; \backslash theta\_\backslash mu\; p\; \backslash ge\; \backslash theta\_\backslash mu\; q.$

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:{{sfn|Ferro|Gerdt|2003|p=83|ps=none}}

• Orderly ranking: $\backslash forall\; y\_i,\; y\_j\; \backslash in\; Y,\; \backslash \; \backslash forall\; \backslash theta\_\backslash mu,\; \backslash theta\_\backslash nu\; \backslash in\; \backslash Theta\; \backslash \; :\; \backslash \; \backslash operatorname\{ord\}(\backslash theta\_\backslash mu)\; \backslash ge\; \backslash operatorname\{ord\}(\backslash theta\_\backslash nu)\; \backslash Rightarrow\; \backslash theta\_\backslash mu\; y\_i\; \backslash ge\; \backslash theta\_\backslash nu\; y\_j$
• Elimination ranking: $\backslash forall\; y\_i,\; y\_j\; \backslash in\; Y,\; \backslash \; \backslash forall\; \backslash theta\_\backslash mu,\; \backslash theta\_\backslash nu\; \backslash in\; \backslash Theta\; \backslash \; :\; \backslash \; y\_i\; \backslash ge\; y\_j\; \backslash Rightarrow\; \backslash theta\_\backslash mu\; y\_i\; \backslash ge\; \backslash theta\_\backslash nu\; y\_j$

In this example, the integer tuple identifies the differential indeterminate and derivative's multi-index, and lexicographic monomial order, $\backslash ge\_\backslash text\{lex\}$, determines the derivative's rank.{{sfn|Wu|2005a|p=4|ps=none}}

: $\backslash eta(\backslash delta\_1^\{e\_1\}\; \backslash circ\; \backslash cdots\; \backslash circ\; \backslash delta\_n^\{e\_n\}(y\_j))=\; (j,\; e\_1,\; \backslash ldots,\; e\_n)$.

: $\backslash eta(\backslash theta\_\backslash mu\; y\_j)\; \backslash ge\_\backslash text\{lex\}\; \backslash eta(\backslash theta\_\backslash nu\; y\_k)\; \backslash Rightarrow\; \backslash theta\_\backslash mu\; y\_j\; \backslash ge\; \backslash theta\_\backslash nu\; y\_k.$

This is the standard polynomial form: $p\; =\; a\_d\; \backslash cdot\; u\_p^d+\; a\_\{d-1\}\; \backslash cdot\; u\_p^\{d-1\}\; +\; \backslash cdots\; +a\_1\; \backslash cdot\; u\_p+\; a\_0$.{{sfn|Kolchin|1973|pp=75–76|ps=none}}{{sfn|Wu|2005a|p=4|ps=none}}

• Leader or leading derivative is the polynomial's highest ranked derivative: $u\_p$.
• Coefficients $a\_d,\; \backslash ldots,\; a\_0$ do not contain the leading derivative $u\_p$.
• Degree of polynomial is the leading derivative's greatest exponent: $\backslash deg\_\{u\_p\}(p)\; =\; d$.
• Initial is the coefficient: $I\_p=a\_d$.
• Rank is the leading derivative raised to the polynomial's degree: $u\_p^d$.
• Separant is the derivative: $S\_p=\; \backslash frac\{\backslash partial\; p\}\{\backslash partial\; u\_p\}$.

Separant set is $S\_A=\; \backslash \{\; S\_p\; \backslash mid\; p\; \backslash in\; A\; \backslash \}$, initial set is $I\_A=\; \backslash \{\; I\_p\; \backslash mid\; p\; \backslash in\; A\; \backslash \}$ and combined set is $H\_A=\; S\_A\; \backslash cup\; I\_A$.{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=159|ps=none}}

Partially reduced (partial normal form) polynomial $q$ with respect to polynomial $p$ indicates these polynomials are non-ground field elements, $p,q\; \backslash in\; \backslash mathcal\{K\}\; \backslash \{\; Y\; \backslash \}\; \backslash setminus\; \backslash mathcal\{K\}$, and $q$ contains no proper derivative of $u\_p$.{{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 $q$ with respect to polynomial $p$ becomes

reduced (normal form) polynomial $q$ with respect to $p$ if the degree of $u\_p$ in $q$ is less than the degree of $u\_\{p\}$ in $p$.{{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 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.{{sfn|Sit|2002|p=6|ps=none}}{{sfn|Kolchin|1973|p=75|ps=none}}

Ritt's reduction algorithm identifies integers $i\_\{A\_\{k\}\},\; s\_\{A\_\{k\}\}$ and transforms a differential polynomial $f$ using pseudodivision to a lower or equally ranked remainder polynomial $f\_\{red\}$ that is reduced with respect to the autoreduced polynomial set $A$. 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:{{sfn|Kolchin|1973|p=75|ps=none}}

: $f\_\backslash text\{red\}\; \backslash equiv\; \backslash prod\_\{A\_k\; \backslash in\; A\}\; I\_\{A\_k\}^\{i\_\{A\_k\}\}\; \backslash cdot\; S\_\{A\_k\}^\{i\_\{A\_k\}\}\; \backslash cdot\; f,\; \backslash pmod\{[A]\}\; \backslash text\{\; with\; \}\; i\_\{A\_k\},\; s\_\{A\_k\}\; \backslash in\; \backslash mathbb\{N\}.$

Set $A$ is a differential chain if the rank of the leading derivatives is and $\backslash forall\; i,\; \backslash \; A\_\{i\}$ is reduced with respect to $A\_\{i+1\}${{sfn|Li|Yuan|2019|p=294|ps=none}}

Autoreduced sets $A$ and $B$ 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}}

• and and $i,j,k\; \backslash in\; \backslash mathbb\{N\}$.
• if there is a $k\; \backslash le\; \backslash operatorname\{minimum\}(m,n)$ such that $A\_i\; =\; B\_i$ for and .
• if and $A\_i\; =\; B\_i$ for $1\; \backslash le\; i\; \backslash le\; n$.
• $\backslash operatorname\{rank\}\; A\; =\; \backslash operatorname\{rank\}\; B$ if $n\; =\; m$ and $A\_i\; =\; B\_i$ for $1\; \backslash le\; i\; \backslash le\; n$.

A characteristic set $C$ is the lowest ranked autoreduced subset among all the ideal's autoreduced subsets whose subset polynomial separants are non-members of the ideal $\backslash mathcal\{I\}$.{{sfn|Kolchin|1973|p=82|ps=none}}

The delta polynomial applies to polynomial pair $p,q$ whose leaders share a common derivative, $\backslash theta\_\{\backslash alpha\}\; u\_\{p\}=\; \backslash theta\_\{\backslash beta\}\; u\_\{q\}$. The least common derivative operator for the polynomial pair's leading derivatives is $\backslash theta\_\{pq\}$, and the delta polynomial is:{{sfn|Kolchin|1973|p=136|ps=none}}{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=160|ps=none}}

: $\backslash operatorname\{\backslash Delta\; -\; poly\}(p,q)=\; S\_\{q\}\; \backslash cdot\; \backslash frac\{\backslash theta\_\{pq\}\; p\}\{\backslash theta\_\{p\}\}\; -\; S\_\{p\}\; \backslash cdot\; \backslash frac\{\backslash theta\_\{pq\}\; q\}\{\backslash theta\_\{q\}\}$

A coherent set 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}}

A regular system $\backslash Omega$ contains a autoreduced and coherent set of differential equations $A$ and a inequation set $H\_\{\backslash Omega\}\; \backslash supseteq\; H\_A$ with set $H\_\backslash Omega$ reduced with respect to the equation set.{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=160|ps=none}}

Regular differential ideal $\backslash mathcal\{I\}\_\backslash text\{dif\}$ and regular algebraic ideal $\backslash mathcal\{I\}\_\backslash text\{alg\}$ are saturation ideals that arise from a regular system.{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=160|ps=none}} Lazard's lemma states that the regular differential and regular algebraic ideals are radical ideals.{{sfn|Morrison|1999|ps=none}}

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

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.{{sfn|Boulier|Lazard|Ollivier|Petitot|1995|p=158|ps=none}}

The membership problem is to determine if a differential polynomial $p$ is a member of an ideal generated from a set of differential polynomials $S$. 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.{{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}}

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

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

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

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

The Pincherle derivative, a derivation of shift-invariant operator $T$, is $T^\{\backslash prime\}\; =\; T\; \backslash circ\; y\; -\; y\; \backslash circ\; T$.{{sfn|Rota|Kahaner|Odlyzko|1973|p=694|ps=none}}

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

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

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

• Every rational number is a quotient of integers.
• : $\backslash forall\; r\; \backslash in\; \backslash mathbb\{Q\},\; \backslash \; \backslash exists\; \backslash \; a\; \backslash in\; \backslash mathbb\{Z\},\; \backslash \; b\; \backslash in\; \backslash mathbb\{Z\}/\; \backslash \{\; 0\; \backslash \},\; \backslash \; r=\backslash frac\{a\}\{b\}$
• Apply the derivation formula for quotients recognizing that derivations of integers are zero:
• : $\backslash delta\; (r)=\; \backslash delta\; \backslash left\; (\; \backslash frac\{a\}\{b\}\; \backslash right\; )\; =\; \backslash frac\{\backslash delta(a)\; \backslash cdot\; b\; -\; a\; \backslash cdot\; \backslash delta(b)\}\{b^\{2\}\}=0$.

Constants form the subring of constants $(\backslash mathbb\{C\},\; \backslash partial\_\{y\})\; \backslash subset\; (\backslash mathbb\{C\}\; \backslash \{\; y\; \backslash \},\; \backslash partial\_\{y\})$.{{sfn|Kolchin|1973|p=60|ps=none}}

Element $\backslash exp(y)$ simply generates differential ideal $[\backslash exp(y)]$ in the differential ring $(\backslash mathbb\{C\}\; \backslash \{\; y,\; \backslash exp(y)\; \backslash \},\; \backslash partial\_\{y\})$

.{{sfn|Sit|2002|p=4|ps=none}}

Any ring with identity is a $\backslash operatorname\{\backslash mathcal\{Z\}-\}$algebra.{{sfn|Dummit|Foote|2004|p=343|ps=none}} Thus a differential ring is a $\backslash operatorname\{\backslash mathcal\{Z\}-\}$algebra.

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

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

: $\backslash partial\_y(y)=1,\; \backslash \; \backslash partial\_y(z)=1+z^2,\; \backslash \; z=\backslash tan(y)$

: $p(y)=1+y^2,\; \backslash \; \backslash partial\_y(p)=2\; \backslash cdot\; y,\backslash \; \backslash gcd(p,\; \backslash partial\_y(p))=1$

: $q(z)=1+z^2,\; \backslash \; \backslash partial\_y(q)=2\; \backslash cdot\; z\; \backslash cdot\; (1+z^2),\backslash \; \backslash gcd(q,\; \backslash partial\_\{y\}(q))=q$

Ring $(\backslash mathbb\{Q\}\; \backslash \{\; y\_\{1\},\; y\_\{2\}\; \backslash \},\; \backslash delta)$ has derivatives $\backslash delta(y\_\{1\})=y\_\{1\}^\{\backslash prime\}$ and $\backslash delta(y\_\{2\})=y\_\{2\}^\{\backslash prime\}$

• Map each derivative to an integer tuple: $\backslash eta(\; \backslash delta^\{(i\_\{2\})\}(y\_\{i\_\{1\}\})\; )=(i\_\{1\},\; i\_\{2\})$.
• Rank derivatives and integer tuples: $y\_\{2\}^\{\backslash prime\; \backslash prime\}\; \backslash \; (2,2)\; >\; y\_\{2\}^\{\backslash prime\}\; \backslash \; (2,1)\; >\; y\_\{2\}\; \backslash \; (2,0)\; >\; y\_\{1\}^\{\backslash prime\; \backslash prime\}\; \backslash \; (1,2)\; >\; y\_\{1\}^\{\backslash prime\}\; \backslash \; (1,1)\; >\; y\_\{1\}\; \backslash \; (1,0)$.

The leading derivatives, and initials are:

: $p=\{\backslash color\{Blue\}\; (y\_\{1\}+\; y\_\{1\}^\{\backslash prime\})\}\; \backslash cdot\; (\{\backslash color\{Red\}\; y\_\{2\}^\{\backslash prime\; \backslash prime\}\})^\{2\}\; +\; 3\; \backslash cdot\; y\_\{1\}^\{2\}\; \backslash cdot\; \{\backslash color\{Red\}y\_\{2\}^\{\backslash prime\; \backslash prime\}\}\; +\; (y\_\{1\}^\{\backslash prime\})^\{2\}$

: $q=\{\backslash color\{Blue\}(y\_\{1\}+\; 3\; \backslash cdot\; y\_\{1\}^\{\backslash prime\})\}\; \backslash cdot\; \{\backslash color\{Red\}\; y\_\{2\}^\{\backslash prime\; \backslash prime\}\}\; +\; y\_\{1\}\; \backslash cdot\; y\_\{2\}^\{\backslash prime\}\; +\; (y\_\{1\}^\{\backslash prime\})^\{2\}$

: $r=\; \{\backslash color\{Blue\}\; (y\_\{1\}+3)\}\; \backslash cdot\; (\{\backslash color\{Red\}\; y\_\{1\}^\{\backslash prime\; \backslash prime\}\})^\{2\}\; +\; y\_\{1\}^\{2\}\; \backslash cdot\; \{\backslash color\{Red\}\; y\_\{1\}^\{\backslash prime\; \backslash prime\}\}+\; 2\; \backslash cdot\; y\_\{1\}$

: $S\_\{p\}=\; 2\; \backslash cdot\; (y\_\{1\}+\; y\_\{1\}^\{\backslash prime\})\; \backslash cdot\; y\_\{2\}^\{\backslash prime\; \backslash prime\}\; +\; 3\; \backslash cdot\; y\_\{1\}^\{2\}$.

: $S\_\{q\}=\; y\_\{1\}+\; 3\; \backslash cdot\; y\_\{1\}^\{\backslash prime\}$

: $S\_\{r\}=\; 2\; \backslash cdot\; (y\_\{1\}+3)\; \backslash cdot\; y\_\{1\}^\{\backslash prime\; \backslash prime\}\; +\; y\_\{1\}^\{2\}$

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

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 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.{{sfn|Hubert|2002|pp=41–47|ps=none}}

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.{{sfn|Harrington|VanGorder|2017|ps=none}} Researchers have applied differential elimination to understanding cellular biology, compartmental biochemical models, parameter estimation and 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 properties of 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}}

A $\backslash operatorname\{\backslash mathbb\{Z\}-graded\}$ vector space $V\_\{\backslash bullet\}$ is a collection of vector spaces $V\_\{m\}$ with integer degree $|v|=m$ for $v\backslash in\; V\_\{m\}$. A direct sum can represent this graded vector space:{{sfn|Keller|2019|p=48|ps=none}}

: $V\_\{\backslash bullet\}\; =\; \backslash bigoplus\_\{m\; \backslash in\; \backslash mathbb\{Z\}\}\; V\_\{m\}$

A differential graded vector space or chain complex, is a graded vector space $V\_\{\backslash bullet\}$ with a differential map or boundary map $d\_\{m\}:\; V\_\{m\}\; \backslash to\; V\_\{m-1\}$ with $d\_\{m\}\; \backslash circ\; d\_\{m+1\}\; =\; 0$ .{{sfn|Keller|2019|pp=50–51|ps=none}}

A cochain complex is a graded vector space $V^\{\backslash bullet\}$ with a differential map or coboundary map

$d\_\{m\}:\; V\_\{m\}\; \backslash to\; V\_\{m+1\}$ with $d\_\{m+1\}\; \backslash circ\; d\_\{m\}\; =\; 0$.{{sfn|Keller|2019|pp=50–51|ps=none}}

A differential graded algebra is a graded algebra $A$ with a linear derivation $d:\; A\; \backslash to\; A$ with $d\; \backslash circ\; d=0$ that follows the graded Leibniz product rule.{{sfn|Keller|2019|pp=58–59|ps=none}}

• Graded Leibniz product rule: $\backslash forall\; a,b\; \backslash in\; A,\; \backslash \; d(a\; \backslash cdot\; b)=d(a)\; \backslash cdot\; b\; +\; (-1)^$

  a

  \cdot a \cdot d(b) with $|a|$ the degree of vector $a$.

A Lie algebra is a finite-dimensional real or complex vector space $\backslash mathcal\{g\}$ with a bilinear bracket operator $[,]:\backslash mathcal\{g\}\; \backslash times\; \backslash mathcal\{g\}\; \backslash to\; \backslash mathcal\{g\}$ with Skew symmetry and the Jacobi identity property.{{sfn|Hall|2015|p=49|ps=none}}

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

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

The adjoint operator, $\backslash operatorname\{ad\_\{X\}\}(Y)=[Y,X]$ 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 $X$.{{sfn|Hall|2015|p=51|ps=none}}{{sfn|Jacobson|1979|p=9|ps=none}}

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

The universal enveloping algebra $U(\backslash mathcal\{g\})$ of Lie algebra $\backslash mathcal\{g\}$ is a maximal associative algebra with identity, generated by Lie algebra elements $\backslash mathcal\{g\}$ 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.{{sfn|Hall|2015|p=247|ps=none}}

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

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

The Weyl algebra is an algebra $A\_\{n\}(K)$ over a ring $K\; [p\_\{1\},\; q\_\{1\},\; \backslash dots,\; p\_\{n\},\; q\_\{n\}]$ with a specific noncommutative product: {{sfn|Lam|1991|pp=7–8|ps=none}}

: $p\_\{i\}\; \backslash cdot\; q\_\{i\}\; -\; q\_\{i\}\; \backslash cdot\; p\_\{i\}=1,\; \backslash \; :\; \backslash \; i\; \backslash in\; \backslash \{1,\; \backslash dots,\; n\; \backslash \}$.

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

: $p\_\{i\}\; \backslash cdot\; q\_\{j\}\; -\; q\_\{j\}\; \backslash cdot\; p\_\{i\}=0\; \backslash text\{\; if\; \}\; i\; \backslash ne\; j,\; \backslash \; p\_\{i\}\; \backslash cdot\; p\_\{j\}\; -\; p\_\{j\}\; \backslash cdot\; p\_\{i\}=0,\; \backslash \; q\_\{i\}\; \backslash cdot\; q\_\{j\}\; -\; q\_\{j\}\; \backslash cdot\; q\_\{i\}=0$.

A Weyl algebra can represent the derivations for a commutative ring's polynomials $f\; \backslash in\; K[y\_\{1\},\; \backslash ldots,\; y\_\{n\}]$. The Weyl algebra's elements are endomorphisms, the elements $p\_\{1\},\; \backslash ldots,\; p\_\{n\}$ function as standard derivations, and map compositions generate linear differential operators. D-module is a related approach for understanding differential operators. The endomorphisms are:{{sfn|Lam|1991|pp=7–8|ps=none}}

: $q\_\{j\}\; (y\_\{k\})=\; y\_\{j\}\; \backslash cdot\; y\_\{k\},\; \backslash \; q\_\{j\}(c)=\; c\; \backslash cdot\; y\_\{j\}\; \backslash text\{\; with\; \}\; c\; \backslash in\; K,\; \backslash \; p\_\{j\}(y\_\{j\})=1,\; \backslash \; p\_\{j\}(y\_\{k\})=0\; \backslash text\{\; if\; \}\; j\; \backslash ne\; k,\; \backslash \; p\_\{j\}(c)=\; 0\; \backslash text\{\; with\; \}\; c\; \backslash in\; K$

The associative, possibly noncommutative ring $A$ has derivation $d:\; A\; \backslash to\; A$.{{sfn|Parshin|1999|p=268|ps=none}}

The pseudo-differential operator ring $A((\backslash partial^\{-1\}))$ is a left $\backslash operatorname\{A-module\}$

containing ring elements $L$:{{sfn|Parshin|1999|p=268|ps=none}}{{sfn|Dummit|Foote|2004|p=337|ps=none}}{{sfn|Taylor|1991|ps=none}}

: $a\_i\; \backslash in\; A,\; \backslash \; i,i\_\{\backslash min\}\; \backslash in\; \backslash mathbb\{N\},\; \backslash \; |i\_\{\backslash min\}|\; >\; 0\; \backslash \; :\; \backslash \; L=\; \backslash sum\_\{i\; \backslash ge\; i\_\{\backslash min\}\}^n\; a\_i\; \backslash cdot\; \backslash partial^i$

The derivative operator is $d(a)\; =\; \backslash partial\; \backslash circ\; a\; -\; a\; \backslash circ\; \backslash partial$.{{sfn|Parshin|1999|p=268|ps=none}}

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

Pseudo-differential operator multiplication is:{{sfn|Parshin|1999|p=268|ps=none}}

: $\backslash sum\_\{i\; \backslash ge\; i\_\{\backslash min\}\}^n\; a\_i\; \backslash cdot\; \backslash partial^i\; \backslash cdot\; \backslash sum\_\{j\backslash ge\; j\_\{\backslash min\}\}^m\; b\_\{i\}\; \backslash cdot\; \backslash partial^j\; =\; \backslash sum\_\{i,j;k\; \backslash ge\; 0\}\; \backslash Bigl(\; \{i\; \backslash atop\; k\}\; \backslash Bigr)\; \backslash cdot\; a\_i\; \backslash cdot\; d^k(b\_j)\; \backslash cdot\; \backslash partial^\{i+j-k\}$

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.{{sfn|Golubitsky|Kondratieva|Ovchinnikov|2009|ps=none}}

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

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

• {{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

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• {{cite book |last1=Wu |first1=Wen-tsün |title=Differential equations with symbolic computation |chapter=On “Good” Bases of Algebraic-Differential Ideals |date=2005a |publisher=Birkhäuser |isbn=978-3-7643-7429-7 |pages=343–350

|doi=10.1007/3-7643-7429-2_19 |chapter-url=https://link.springer.com/chapter/10.1007/3-7643-7429-2_19 |language=en}}

• {{cite book |last1=Wu |first1=Wen-tsün |title=Differential equations with symbolic computation |chapter=On the Construction of Groebner Basis of a Polynomial Ideal Based on Riquier–Janet Theory |series=Trends in Mathematics |date=2005b |publisher=Birkhäuser |isbn=978-3-7643-7429-7 |pages=351–368 |doi=10.1007/3-7643-7429-2_20 |chapter-url=//link.springer.com/chapter/10.1007/3-7643-7429-2_20 |language=en}}
• {{cite journal |last1=Zharinov |first1=V. V. |title=Navier–Stokes equations, the algebraic aspect |journal=Theoretical and Mathematical Physics |date=December 2021 |volume=209 |issue=3 |pages=1657–1672 |doi=10.1134/S0040577921120011 |arxiv=2110.01504 |bibcode=2021TMP...209.1657Z |s2cid=238259977 |url=https://link.springer.com/content/pdf/10.1134/S0040577921120011.pdf}}

{{refend}}

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