Английская Википедия:Differential algebra

In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras and Lie algebras may be considered as belonging to differential algebra.

More specifically, differential algebra refers to the theory introduced by Joseph Ritt in 1950, in which differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations.Шаблон:Sfn Шаблон:Sfn Шаблон:Sfn

A natural example of a differential field is the field of rational functions in one variable over the complex numbers, <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.Шаблон:Sfn 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.Шаблон:Sfn Шаблон:Sfn Шаблон:Sfn Ellis Kolchin, Ritt's student, advanced this field and published Differential Algebra And Algebraic Groups.Шаблон:Sfn

Differential rings

Definition

A derivation <math display="inline"> \partial </math> on a ring <math display="inline"> \mathcal{R} </math> is a 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> (Leibniz product rule),

for every <math>r_1</math> and <math>r_2</math> in <math>R.</math>

A derivation is 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>Шаблон:Sfn When there is only one derivation one talks often of an ordinary differential ring; otherwise, one talks of a partial differential ring.

A differential field is differentiable ring that is also a field. A differential algebra <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 constants 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.Шаблон:Sfn This meaning of "constant" generalizes the concept of a constant function, and must not be confused with the common meaning of a constant.

Basic formulas

In the following identities, <math>\delta</math> is a derivation of a differential ring <math>R.</math>Шаблон:Sfn

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 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 logarithmic derivative identity: <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 derivation operator or higher-order derivationШаблон:Citation needed is the composition of several derivations. As the derivations of a differential ring are supposed to commute, the order of the derivations does not matter, and a derivation operator may be written as <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 proper derivative is a derivative of positive order.Шаблон:Sfn

Differential ideals

A differential ideal <math>I</math> of a differential ring <math>R</math> is an ideal of the ring <math>R</math> that is 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 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.Шаблон:Sfn 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.Шаблон:Sfn 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.Шаблон:Sfn Шаблон:Sfn

The algebraic ideal generated by <math>S</math> is the set of the 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 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 fractions <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 the differential polynomials 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 Шаблон:Math. 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. This makes the theory of this generalization of polynomial rings difficult. However, two facts allow such a generalization.

Firstly, a finite number of differential polynomial 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 <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 Ritt-Raudenbush basis theorem which asserts that if <math>R</math> is a Ritt Algebra (that, is a differential ring containing the field of rational numbers),Шаблон:Sfn 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).Шаблон:Sfn Шаблон:Sfn

This Noetherian property implies that, in a ring of differential polynomial, every radical differential ideal is finitely generated, in the sense that it is the smallest radical differential ideal that contains a finite set of polynomials.Шаблон:Sfn 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.Шаблон:Sfn

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.Шаблон:Sfn Шаблон:Sfn Шаблон:Sfn Шаблон:Sfn Шаблон:Sfn Шаблон:Sfn Шаблон:Sfn

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:Шаблон:Sfn Шаблон:Sfn Шаблон:Sfn

<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:Шаблон:Sfn

Orderly ranking: <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>
Elimination ranking: <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 monomial order, <math display="inline"> \ge_\text{lex}</math>, determines the derivative's rank.Шаблон:Sfn

<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>.Шаблон:Sfn Шаблон:Sfn

Leader or leading derivative is the polynomial's highest ranked derivative: <math>u_p</math>.
Coefficients <math>a_d, \ldots, a_0</math> do not contain the leading derivative <math display="inline">u_p</math>.
Degree of polynomial is the leading derivative's greatest exponent: <math>\deg_{u_p}(p) = d</math>.
Initial is the coefficient: <math> I_p=a_d</math>.
Rank is the leading derivative raised to the polynomial's degree: <math>u_p^d</math>.
Separant 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>.Шаблон:Sfn

Reduction

Partially reduced (partial normal form) 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>.Шаблон:Sfn Шаблон:Sfn Шаблон:Sfn

Partially reduced polynomial <math display="inline">q</math> with respect to polynomial <math display="inline">p</math> becomes reduced (normal form) 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>.Шаблон:Sfn Шаблон:Sfn Шаблон:Sfn

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.Шаблон:Sfn Шаблон:Sfn

Ritt's reduction algorithm identifies integers <math display="inline">i_{A_{k}}, s_{A_{k}}</math> and transforms a differential polynomial <math display="inline">f</math> using 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:Шаблон:Sfn

<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 differential chain 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>Шаблон:Sfn

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.Шаблон:Sfn

<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 characteristic set <math display="inline">C</math> is the 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>.Шаблон:Sfn

The delta polynomial 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:Шаблон:Sfn Шаблон:Sfn

<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 coherent set is a polynomial set that reduces its delta polynomial pairs to zero.Шаблон:Sfn Шаблон:Sfn

Regular system and regular ideal

A regular system <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.Шаблон:Sfn

Regular differential ideal <math display="inline">\mathcal{I}_\text{dif} </math> and regular algebraic ideal <math display="inline">\mathcal{I}_\text{alg} </math> are saturation ideals that arise from a regular system.Шаблон:Sfn Lazard's lemma states that the regular differential and regular algebraic ideals are radical ideals.Шаблон:Sfn

Regular differential ideal: <math display="inline">\mathcal{I}_\text{dif}=[A]:H_\Omega^\infty.</math>
Regular algebraic ideal: <math display="inline">\mathcal{I}_\text{dif}=(A):H_\Omega^\infty.</math>

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.Шаблон:Sfn

The membership problem 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.Шаблон:Sfn

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

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 standard derivation.

Example 2: <math display="inline">(\mathbb{C} \{ y \}, (1+3 \cdot y + y^{2}) \cdot \partial_{y} ) </math> is a differential field with a linear differential operator as the derivation.

Derivation

Define <math display="inline">E^{a}(p(y))=p(y+a)</math> as shift operator <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 Pincherle derivative, a derivation of shift-invariant operator <math display="inline">T</math>, is <math display="inline">T^{\prime} = T \circ y - y \circ T </math>.Шаблон:Sfn

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 subring of constants <math display="inline">(\mathbb{C}, \partial_{y}) \subset (\mathbb{C} \{ y \}, \partial_{y}) </math>.Шаблон:Sfn

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>.Шаблон:Sfn

Algebra over a differential ring

Any ring with identity is a <math display="inline">\operatorname{\mathcal{Z}-}</math>algebra.Шаблон:Sfn 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.Шаблон:Sfn Thus, a differential ring is an algebra over its differential subring. This is the natural structure of an algebra over its subring.Шаблон:Sfn

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 leading derivatives, and initials 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.Шаблон:Sfn

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.Шаблон:Sfn

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

Algebras with derivations

Differential graded vector space

A <math display="inline">\operatorname{\mathbb{Z}-graded}</math> vector space <math display="inline">V_{\bullet} </math> is a collection of vector spaces <math display="inline">V_{m}</math> with integer degree <math display="inline">|v|=m</math> for <math display="inline"> v\in V_{m}</math>. A direct sum can represent this graded vector space:Шаблон:Sfn

<math>V_{\bullet} = \bigoplus_{m \in \mathbb{Z}} V_{m}</math>

A differential graded vector space or chain complex, is a graded vector space <math display="inline">V_{\bullet}</math> with a differential map or boundary map <math display="inline">d_{m}: V_{m} \to V_{m-1}</math> with <math> d_{m} \circ d_{m+1} = 0 </math> .Шаблон:Sfn

A cochain complex is a graded vector space <math display="inline">V^{\bullet}</math> with a differential map or coboundary map <math display="inline">d_{m}: V_{m} \to V_{m+1}</math> with <math> d_{m+1} \circ d_{m} = 0 </math>.Шаблон:Sfn

Differential graded algebra

A differential graded algebra 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.Шаблон:Sfn

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 Lie algebra is a finite-dimensional real or complex vector space <math display="inline">\mathcal{g}</math> with a bilinear bracket operator <math display="inline">[,]:\mathcal{g} \times \mathcal{g} \to \mathcal{g} </math> with Skew symmetry and the Jacobi identity property.Шаблон:Sfn

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 adjoint operator, <math display="inline">\operatorname{ad_{X}}(Y)=[Y,X]</math> 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 <math display="inline">X</math>.Шаблон:Sfn Шаблон:Sfn

<math> \operatorname{ad}_{X}([Y,Z]) = [\operatorname{ad}_{X}(Y),Z] + [Y,\operatorname{ad}_{X}(Z)] </math>

The universal enveloping algebra <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.Шаблон:Sfn

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: Шаблон:Sfn

<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 endomorphisms, the elements <math display="inline">p_{1}, \ldots, p_{n}</math> function as standard derivations, and map compositions generate linear differential operators. D-module is a related approach for understanding differential operators. The endomorphisms are:Шаблон:Sfn

Pseudodifferential operator ring

The associative, possibly noncommutative ring <math display="inline">A</math> has derivation <math display="inline">d: A \to A </math>.Шаблон:Sfn

The pseudo-differential operator ring <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>:Шаблон:Sfn Шаблон:Sfn Шаблон:Sfn

<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>.Шаблон:Sfn

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

Pseudo-differential operator multiplication is:Шаблон:Sfn

<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 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.Шаблон:Sfn

The Kolchin catenary conjecture 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.Шаблон:Sfn

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.Шаблон:Sfn

Citations

Шаблон:Reflist

References

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External links

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

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