User:TMM53/tmp/differential algebra 2023-03-06a

In mathematics, differential rings, differential fields and differential algebras are rings, fields and algebras equipped with a finite set of derivations. Differential algebra includes the study of these algebraic objects and their use in the algebraic study of differential equations. This approach provides an improved understanding in many areas of mathematics including algebraic geometry, differential equations and symbolic integration. Direct applications have occurred in many areas including chemical engineering, computational biolology, control theory and theoretical physics.

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. His efforts led to an initial paper Manifolds Of Functions Defined By Systems Of Algebraic Differential Equations and 2 books, Differential Equations From The Algebraic Standpoint and Differential Algebra. Ellis Kolchin, Ritt's student, advanced this field and published Differential Algebra And Algebraic Groups.

Differential ring
A derivation $ \delta $  on a ring $ \mathcal{R} $  is a linear unary operator and an additive group homomorphism that follows an addition rule and Leibniz product rule, $ \forall r_{1},r_{2} \in \mathcal{R}$ :
 * $ \delta (r_{1}+r_{2})=\delta (r_{1})+\delta (r_{2}) $
 * $ \delta (r_{1} \cdot r_{2}) = \delta (r_{1}) \cdot r_{2} + r_{1} \cdot \delta (r_{2}) $

A differential ring is a commutative ring $ \mathcal{R} $  with a multiplicative  identity of 1 (unital ring) and a finite set of commutative derivations  $ \Delta = \{ \delta_{1}, \ldots, \delta_{n} \} $  that  map ring elements to ring elements, $ \Delta: \mathcal{R} \to \mathcal{R} $. An ordinary differential ring's derivation set contains one derivation; a partial differential ring's derivation set contains multiple derivations. Abbreviated notations are $\operatorname{\Delta - \mathcal{R}}$ or $(\mathcal{R}, \Delta)$ for partial differential rings and $ \operatorname{\delta - \mathcal{R}}$  or $(\mathcal{R}, \delta)$ for ordinary differential rings. The constants set $\mathcal{Const}_{\Delta}(\mathcal{R})$ contains ring elements that every derivation maps to zero.

Derivation formulas
Some derivations formulas apply to a differential field or a differential integral domain.
 * $$ \delta (c \cdot r)= c \cdot \delta (r), \ r \in \mathcal{R}, \ c \in \mathcal{Const}_{\delta} ( \mathcal{R} ) $$
 * $$ \delta \left( \frac{r_{1}}{r_{2}} \right)= \frac{\delta (r_{1}) \cdot r_{2} - r_{1} \cdot \delta (r_{2})}{r_{2}^{2}}, \ r_{2} \ne 0, \ r_{1},r_{2} \in \mathcal{R}, \ \mathcal{R} \text{ is a field} $$
 * $$ \delta (r^{n})= n \cdot r^{n-1} \cdot \delta (r), \ r \in \mathcal{R} \backslash \{0 \}, \ n \text{ is a positive integer.} $$
 * $$ \frac{\delta (u_{1}^{e_{1}} \ldots u_{n}^{e_{n}})}{(u_{1}^{e_{1}} \ldots u_{n}^{e_{n}})} = e_{1} \frac{\delta( u_{1} ) }{u_{1}} + \dots + e_{n} \frac{\delta( u_{n} ) }{u_{n}}, \text{ } u_{1}, \dots  u_{n} \in $$ Units of $$ \mathcal{R}, \ e_{1}, \ldots, e_{n} \text{ are integers }, \mathcal{R} \text{ is an integral domain.} $$

The last formula is the logarithmic derivative identity.

Derivative operator
The derivative operator is a sequence of composed derivations, each derivation occurring one or multiple times. An integer superscript indicates the number of derivations for partial differential rings, and superscript primes indicate the number of derivations for ordinary differential rings. Proper derivatives contains at least one derivation. Derivative operators form a free commutative semigroup generated by the derivation set. The multi-index, an integer tuple, identifies the number of derivations from each derivation operator. The order of the derivative operator is the total number of derivations. A derivative is the application of a derivative operator to a set element.
 * Derivative operator: $$ \theta_{\mu} = \delta_{1}^{e_{1}} \circ \ldots \circ \delta_{n}^{e_{n}}$$.
 * Derivative multi-index: $$ \mu=(e_{1}, \dots, e_{n})$$.
 * Order of derivative: $$ \operatorname{ord}(\theta_{\mu}) = | \mu | = e_{1} + \dots + e_{n} $$.
 * Derivative of $$a \in A $$: $$ \ \theta_{\mu} a = \delta_{1}^{e_{1}} \circ \ldots \circ \delta_{n}^{e_{n}}(a)$$.
 * Derivative operator set: $ \Theta = \{ \delta_{1}^{e_{1}} \circ \ldots \circ \delta_{n}^{e_{n}} \ | \ \delta_{i} \in \Delta, \ e_{i} \in \mathbb{N} \} $.
 * Derivative set: $ \Theta A = \{ \theta_{\mu} a \ | \ \theta_{\mu} \in \Theta, a \in A \} $.

Subrings
The $\operatorname{\Delta_{R} - \mathcal{R}}$ is a differential subring of $ \operatorname{\Delta_{S} - \mathcal{S}} $  if $ \mathcal{R}$  is a subring of $ \mathcal{S}$, and the derivation set $ \operatorname{\Delta_{R}} $  is the derivation set $ \operatorname{\Delta_{S}} $  restricted to $ \mathcal{R}$. An equivalent statement is $ \operatorname{\Delta_{S} - \mathcal{S}} $ is the differential  overring of $ \operatorname{\Delta_{R} - \mathcal{R}} $.

The intersection of any family of differential subrings is a differential subring. The intersection of any set of differential subrings containing a common set is a differential subring, and the smallest differential subring containing a common set is the intersection of all subrings containing the common set.

Set $ \Theta A $ generates differential ring $\mathcal{R} \{ A \} $  over $\mathcal{R}$. This is the smallest differential subring containing differential subring $\mathcal{R} $ and set $ \Theta A $. A finitely generated differential subring arises from a finite set, and a simply generated differential subring arises from a single element. Adjoining or adding an element to the generator set extends the differential ring. Using the square bracket notation for ring extension, $\mathcal{R} \{ A \}=\mathcal{R} [ \Theta A ] $.

Set $ \Theta A $ generates differential field $\mathcal{F} \langle A \rangle $  over field $\mathcal{F}$. Using the parentheses notation for a field extension, $\mathcal{F} \langle A \rangle =\mathcal{F} ( \Theta A ) $.

A field $K$ is a closed differential field if each instance when a differential equation set's solution, $f_{i} \in K \{ y_{1}, \ldots y_{m} \}$  for $i \in \{ 1, \ldots, m \}$, occurs in field $L$  extended over $K$ , the  solution occurs in the field $K$. Any differential field may extend to a closed differential field. Differential Galois theory studies differential field extensions and the associated Galois group.

Ideals
A differential ideal of $ \mathcal{R}$  is an ideal closed (stable) under the ring's derivation set $ \mathcal{\Delta}$. A differential proper ideal is a proper subset of the differential ring. The intersection, sum , and finite product of any family of differential ideals is a differential ideal. A radical differential ideal or perfect differential ideal is an ideal equal to its radical: $ \mathcal{I} = \sqrt{\mathcal{I}} $.

The smallest ideal generated from ring $\mathcal{R} $ by a set includes:
 * Ideal generated by set $$A$$: $$ \ (A)_{R} $$
 * Differential ideal generated by set $$ A $$: $$ \ [A]_{R} $$
 * Radical differential ideal generated by set $$A$$: $$ \ \{A \}_{R} $$

Ring homomorphism
A differential ring homomorphism is a map, $ \operatorname{f}: \mathcal{R} \to \mathcal{S} $ of differential rings that share the same derivation set, $ \Delta_{R}=\Delta_{S} $, and the ring homomorphism commutes with derivation, $ \forall r \in \mathcal{R}, \ \forall \delta \in \Delta \ : \ \delta (\operatorname{f}(r))= \operatorname{f}(\delta(r)) $.
 * The kernel is a differential ideal of $ \mathcal{R}$, and the image is a differential subring.
 * The ring $ \mathcal{S}$ is an extension of $ \mathcal{R}$, and $ \mathcal{R}$  is a subring of $ \mathcal{S}$  if the ring homomorphism is an  inclusion.
 * For differential ring $\mathcal{R} $ and differential ideal $\mathcal{I} $, the  canonical homomorphism maps the ring to the differential  residue ring: $ \operatorname{f}: \mathcal{R} \to \mathcal{R} / \mathcal{I} $.

Modules
A differential $ \operatorname{\mathcal{R} - module}$ or module over differential ring $ \operatorname{\Delta - \mathcal{R}} $ has module $ \mathcal{M} $  whose elements follow these sum and product derivation rules: $ \delta \in \Delta, \ r \in \mathcal{R}, \ u,v \in \mathcal{M} $ :
 * $ \delta(u+v)= \delta (u) + \delta (v) $
 * $ \delta(r \cdot u)= \delta (r) \cdot u + r \cdot \delta (u) $

A differential vector space is a differential module over a differential field.

A differential $ \operatorname{\mathcal{R}-algebra}$ or differential algebra over the $ \mathcal{R} $  is the ring $ \mathcal{M} $, the $ \operatorname{\mathcal{R}-algebra}$ , and a derivation set $\Delta$  that makes $ \mathcal{M} $  a differential ring and that follows this derivation product rule:
 * $$ \forall \delta \in \Delta, \ \forall r \in \mathcal{R}, \ \forall u \in \mathcal{M} \ : \ \delta(r \cdot u)= \delta (r) \cdot u + r \cdot \delta (u) $$.

Polynomials
The derivatives $ \Theta Y $ of the set of differential indeterminates $Y$  generate the differential  polynomial ring $ \mathcal{K} \{ Y \}=\mathcal{K} \{ y_{1}, \dots, y_{m} \}$  over the  ground field $ \mathcal{K} $. Unless otherwise noted, polynomial statements assume a characteristic zero.

The standard derivation for ring $ (\mathcal{K} \{ y_{1}, \ldots, y_{m} \}, \Delta = \{ \partial_{1}, \ldots, \partial_{n} \} )$ is
 * $$\partial_{i}(y_{j})=

\begin{cases} 1 & \text{ if } i=j, \\ 0 & \text{ if } i \ne j \end{cases}$$

An algebraically independent differential field $ \mathcal{F} \{ Y \} $ is a differential field with a non-vanishing  Wronskian determinant.

Special and normal polynomials have distinct 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 squarefree. The definitions are:
 * Normal polynomial $$p$$: $$ \ gcd(p,\delta(p))=1$$.
 * Special polynomial $$q$$: $$ \ gcd(q,\delta(q))=q$$.

A Ritt Algebra is a differential ring containing the field of rational numbers.

The Ritt-Raudenbush basis theorem states that if $ \mathcal{K} $ is a Ritt Algebra satisfying the ascending chain condition on radical differential ideals, then the differential ring arising from adjoining a finite number of differential indeterminants, $ \mathcal{K}\{ Y \} $, will satisfy the ascending chain condition on radical differential ideals. Implications are:
 * A radical differential ideal is the radical of a finitely generated ideal.
 * A radical differential ideal is an intersection of a finite set of distinct unique prime ideals called essential prime components.

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.

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:
 * $ \forall p \in \Theta Y, \ \forall \theta_{ \mu } \in \Theta : \theta_{ \mu } p > p $.
 * $ \forall p,q \in \Theta Y, \ \forall \theta_{ \mu } \in \Theta : p \ge q \Rightarrow \theta_{ \mu } p \ge \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:
 * Orderly ranking : $$ \forall y_{i}, y_{j} \in Y, \ \forall \theta_{\mu}, \theta_{\nu} \in \Theta \ : \ \operatorname{ord}(\theta_{\mu}) \ge \operatorname{ord}(\theta_{\nu}) \Rightarrow \theta_{\mu} y_{i} \ge \theta_{\nu} y_{j}$$
 * Elimination ranking : $$\forall y_{i}, y_{j} \in Y, \ \forall \theta_{\mu}, \theta_{\nu} \in \Theta \ : \ y_{i} \ge y_{j} \Rightarrow \theta_{\mu} y_{i} \ge \theta_{\nu} y_{j}$$

In this example, the integer tuple identifies the differential indeterminate and derivative's multi-index, and lexicographic monomial order, $ \ge_{lex}$, determines the derivative's rank.
 * $$\eta(\delta_{1}^{e_{1}} \circ \ldots \circ \delta_{n}^{e_{n}}(y_{j}))= (j, e_{1}, \dots, e_{n}) $$.
 * $$ \eta(\theta_{ \mu } y_{j}) \ge_{lex} \eta(\theta_{\nu} y_{k}) \Rightarrow \theta_{ \mu } y_{j} \ge \theta_{\nu} y_{k} $$.

Leading derivative, initial and separant
This is the standard polynomial form: $$ p = a_{d} \cdot u_{p}^{d}+ a_{d-1} \cdot u_{p}^{d-1} + \dots +a_{1} \cdot u_{p}+ a_{0} $$.
 * Leader or leading derivative is the polynomial's highest ranked derivative: $$u_{p}$$.
 * Coefficients $$a_{d}, \ldots, a_{0}$$ do not contain the leading derivative $u_{p}$.
 * Degree of polynomial is the leading derivative's greatest exponent: $$\operatorname{deg}_{u_{p}}(p)=d$$.
 * 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: $$ S_{p}= \frac{\partial p}{\partial u_{p}}$$.

Separant set is $$S_{A}= \{ S_{p} \ | \ p \in A \} $$, initial set is $$I_{A}= \{ I_{p} \ | \ p \in A \} $$ and combined set is $H_{A}= S_{A} \cup I_{A} $.

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

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 $p$.

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.

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:
 * $$ f_{red} \equiv \prod_{A_{k} \in A} I_{A_{k}}^{i_{A_{k}}} \cdot S_{A_{k}}^{i_{A_{k}}} \cdot f, \text{    } (mod[A]) \text{  with  } i_{A_{k}}, s_{A_{k}} \in \mathbb{N}$$.

Ranking polynomial sets
Set $A$ is a differential chain if the rank of the leading derivatives is $u_{A_{1}} < \dots < u_{A_{m}} $  and $\forall i, \ A_{i}$  is reduced with respect to $A_{i+1}$

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.
 * $$A_{1} < \ldots < A_{m} \in A $$ and $$B_{1} < \ldots < B_{n} \in B $$ and $$ i,j,k \in \mathbb{N}$$.
 * $$ \text{rank } A < \text{rank } B $$ if there is a $$ k \le minimum(m,n) $$ such that $$ A_{i} = B_{i}$$ for $ 1 \le i < k $ and $$ A_{k} < B_{k} $$.
 * $$ \text{rank } A < \text{rank } B $$ if $$ n < m $$ and $$A_{i} = B_{i}$$ for $$1 \le i \le n $$.
 * $$ \text{rank } A = \text{rank } B $$ if $$ n = m $$ and $$A_{i} = B_{i}$$ for $$1 \le i \le n $$.

Polynomial sets
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 $\mathcal{I}$.

The delta polynomial applies to polynomial pair $p,q$ whose leaders share a common derivative, $\theta_{\alpha} u_{p}= \theta_{\beta} u_{q}$. The least common derivative operator for the polynomial pair's leading derivatives is $\theta_{pq}$, and the delta polynomial is:
 * $$\operatorname{\Delta - poly}(p,q)= S_{q} \cdot \frac{\theta_{pq} p}{\theta_{p}} - S_{p} \cdot \frac{\theta_{pq} q}{\theta_{q}} $$

A coherent set is a polynomial set that reduces its delta polynomial pairs to zero.

Regular system and regular ideal
A regular system $\Omega$ contains a autoreduced and coherent set of differential equations $A$  and a inequation set $H_{\Omega} \supseteq H_{A}$  with set $H_{\Omega}$  reduced with respect to the equation set.

Regular differential ideal $\mathcal{I}_{dif} $ and regular algebraic ideal $\mathcal{I}_{alg} $  are saturation ideals that arise from a regular system. Lazard's lemma states that the regular differential and regular algebraic ideals are radical ideals.
 * Regular differential ideal : $\mathcal{I}_{dif}=[A]:H_{\Omega}^{\infty}$.
 * Regular algebraic ideal : $\mathcal{I}_{dif}=(A):H_{\Omega}^{\infty}$.

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.

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.

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

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

Example 2: $(\mathbb{C} \{ y \}, (1+3 \cdot y + y^{2}) \cdot \partial_{y} ) $ is a differential field with a  linear differential operator as the derivation.

Derivation
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: $E^{a} \circ T=T \circ E^{a}$.

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

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

Field of rational numbers is $$(\mathbb{Q}. \delta)$$, and every rational number is a constant.
 * Every rational number is a quotient of integers.
 * $$ \forall r \in \mathbb{Q}, \ \exists \ a \in \mathbb{Z}, \ b \in \mathbb{Z}/ \{ 0 \}, \ r=\frac{a}{b} $$


 * Apply the derivation formula for quotients recognizing that derivations of integers are zero:
 * $$ \delta (r)= \delta \left ( \frac{a}{b} \right ) = \frac{\delta(a) \cdot b - a \cdot \delta(b)}{b^{2}}=0 $$.

Differential subring
Constants form the subring of constants $(\mathbb{C}, \partial_{y}) \subset (\mathbb{C} \{ y \}, \partial_{y}) $.

Differential ideal
Element $\exp(y)$ simply generates differential ideal $ [\exp(y)] $  in the differential ring $(\mathbb{C} \{ y, \exp(y) \}, \partial_{y}) $.

Algebra over a differential ring
Any ring with identity is a $\operatorname{\mathcal{Z}-}$ algebra. Thus a differential ring is a $\operatorname{\mathcal{Z}-}$ algebra.

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

Special and normal polynomials
Ring $(\mathbb{Q} \{ y, z \}, \partial_{y}) $ has irreducible polynomials, $p$  (normal, squarefree) and $q$  (special, ideal generator).
 * $ \partial_{y}(y)=1, \ \partial_{y}(z)=1+z^{2}, \ z=\tan(y)$
 * $p(y)=1+y^{2}, \ \partial_{y}(p)=2 \cdot y, \ \operatorname{gcd}(p, \partial_{y}(p))=1$
 * $q(z)=1+z^{2}, \ \partial_{y}(q)=2 \cdot z \cdot (1+z^{2}), \ \operatorname{gcd}(q, \partial_{y}(q))=q$

Ranking
Ring $(\mathbb{Q} \{ y_{1}, y_{2} \}, \delta)$ has derivatives $\delta(y_{1})=y_{1}^{\prime}$  and $\delta(y_{2})=y_{2}^{\prime}$
 * Map each derivative to an integer tuple: $\eta( \delta^{(i_{2})}(y_{i_{1}}) )=(i_{1}, i_{2})$.
 * Rank derivatives and integer tuples: $ 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 intial
The leading derivatives, and initials are:
 * $ 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} $
 * $ 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} $
 * $ 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

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

Autoreduced sets

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

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.

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.

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 Lypapunov functions.. Researchers have applied differential elimination to understanding cellular biology, compartmental biochemical models, parameter estimation and quasi-steady state approximation (QSSA) for biochemical reactions. Using differential Gröbner bases, researchers have investigated non-classical symmetry properties of non-linear differential equations. Other applications include control theory, model theory, and algebraic geometry. Differential algebra also applies to differential-difference equations.

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

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

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

Differential graded algebra
A differential graded algebra is a graded algebra $A$ with a linear derivation $d: A \to A $  with $$d \circ d=0 $$ that follows the graded Leibniz product rule.
 * Graded Leibniz product rule: $$\forall a,b \in A, \ d(a \cdot b)=d(a) \cdot b + (-1)^{|a|} \cdot a \cdot d(b)$$ with $$|a|$$ the degree of vector $$a$$.

Lie algebra
A Lie algebra is a finite dimensional real or complex vector space $\mathcal{g}$ with a  bilinear bracket operator $[,]:\mathcal{g} \times \mathcal{g} \to \mathcal{g} $  with  Skew symmetry and the  Jacobi identity property.
 * Skew symmetry: $$\forall X, Y \in \mathcal{g}, \ [X,Y]= -[Y,X]$$.
 * Jacobi identity propert: $$\forall X, Y, Z \in \mathcal{g}, \ [X,[Y,Z]]+[Y,[Z,X]] + [Z,[X,Y]]=0 $$.

The adjoint operator, $\operatorname{ad_{X}}(Y)=[Y,X]$ 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 $X$.
 * $$ \operatorname{ad_{X}}([Y,Z]) = [\operatorname{ad_{X}}(Y),Z] + [Y,\operatorname{ad_{X}}(Z)] $$

The universal enveloping algebra $U(\mathcal{g})$  of Lie algebra $\mathcal{g}$  is a maximal associative algebra with identity, generated by Lie algebra elements $\mathcal{g}$  and containing products defined by the bracket operation. Maximal means that a linear homomorphism maps the universal algebra to any other algebra that otherwise has these properties. The adjoint operator is a derivation following the Leibniz product rule.
 * Product in $$U(\mathcal{g})$$ : $$X \cdot Y - Y \cdot X = [X,Y], \ \forall X,Y \in U(\mathcal{g})$$
 * Leibniz product rule: $$ \forall X,Y,Z \in U(\mathcal{g}) \ : \ Ad_{X}( Y \cdot Z)=Ad_{X}(Y) \cdot Z + Y \cdot Ad_{X}(Z)$$.

Weyl algebra
The Weyl algebra is an algebra $A_{n}(K)$  over a ring $K [p_{1}, q_{1}, \dots, p_{n}, q_{n}]$  with a specific noncommutative product:
 * $$ p_{i} \cdot q_{i} - q_{i} \cdot p_{i}=1, \ : \ i \in \{1, \dots, n \} $$.

All other indeterminate products are commutative for $i,j \in \{1, \dots, n \}$ :
 * $$ p_{i} \cdot q_{j} - q_{j} \cdot p_{i}=0 \text{ if  } i \ne j, \ p_{i} \cdot p_{j} - p_{j} \cdot p_{i}=0, \ q_{i} \cdot q_{j} - q_{j} \cdot q_{i}=0 $$.

A Weyl algebra can represent the derivations for a commutative ring's polynomials $f \in K[y_{1}, \ldots, y_{n}]$. The Weyl algebra's elements are endomorphisms, the elements $p_{1}, \ldots, p_{n}$ function as standard derivations, and map compositions generate linear differential operators. D-module is a related approach for understanding differential operators. The endomorphisms are:
 * $$ q_{j} (y_{k})= y_{j} \cdot y_{k}, \ q_{j}(c)= c \cdot y_{j} \text{ with  } c \in K, \ p_{j}(y_{j})=1, \ p_{j}(y_{k})=0 \text{  if  } j \ne k, \ p_{j}(c)= 0 \text{  with  } c \in K $$

A derivative operator $D^{\alpha}$ and a linear differential operator $P(\textbf{y},D)$  are:
 * $$D^{\alpha}=(-i \cdot \partial_{1}^{e_{1}}) \circ \ldots \circ (-i \cdot \partial_{d}^{e_{n}})$$ with $$ \alpha = (e_{1}, \dots, e_{d}) \in \mathbb{N}^{d} $$
 * $$p(\textbf{y},D)= \sum _{|\alpha| \le M} a_{\alpha} (\textbf{y}) \cdot D^{\alpha}$$ with $$\textbf{y}= (y_{1}, \dots, y_{d})$$

Pseudodifferential operator ring
The associative, possibly noncommutative ring $A$ has derivation $d: A \to A $. The pseudo-differential operator ring $A((\partial^{-1}))$ is a left $\operatorname{A-module}$ containing ring elements $L$ :
 * $$ a_{i} \in A, \ i,i_{min} \in \mathbb{N}, \ |i_{min}| > 0 \ : \ L= \sum_{i \ge i_{min}}^{n} a_{i} \cdot \partial^{i}$$

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

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

Pseudo-differential operator multiplication is:
 * $$\sum_{i \ge i_{min}}^{n} a_{i} \cdot \partial^{i} \cdot \sum_{j\ge j_{min}}^{m} b_{i} \cdot \partial^{j} = \sum_{i,j;k \ge 0} \Bigl( {i \atop k} \Bigr) \cdot a_{i} \cdot d^{k}(b_{j}) \cdot \partial^{i+j-k}$$

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

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

The Jacobi bound conjecture concerns the upper bound for the order of an differential variety's irreducible component. The polynomial's orders determine a Jacobi number, and the conjecture is the Jacobi number determines this bound.