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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and
differential operators In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
as algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as
polynomial algebra In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (mathematics), ring formed from the set (mathematics), set of polynomials in one or more indeterminate (variable), indeterminates (traditionally ...
s are used for the study of
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
, which are solution sets of
systems of polynomial equations A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some field . A ''solution'' of a polynomial system is a set of values for the ...
.
Weyl algebra In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. ...
s and
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s may be considered as belonging to differential algebra. More specifically, ''differential algebra'' refers to the theory introduced by Joseph Ritt in 1950, in which differential rings, differential fields, and differential algebras are rings,
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
, and
algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
equipped with finitely many derivations. A natural example of a differential field is the field of
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s in one variable over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s, \mathbb(t), where the derivation is differentiation with respect to t. More generally, every differential equation may be viewed as an element of a differential algebra over the differential field generated by the (known) functions appearing in the equation.


History

Joseph Ritt developed differential algebra because he viewed attempts to reduce systems of differential equations to various canonical forms as an unsatisfactory approach. However, the success of algebraic elimination methods and algebraic manifold theory motivated Ritt to consider a similar approach for differential equations. 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 rings


Definition

A
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a ...
\partial on a ring R is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
\partial : R \to R\, such that \partial(r_1 + r_2) = \partial r_1 + \partial r_2 and : \partial(r_1 r_2) = (\partial r_1) r_2 + r_1 (\partial r_2)\quad ( Leibniz product rule), for every r_1 and r_2 in R. A derivation is
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
over the integers since these identities imply \partial (0)=\partial (1) = 0 and \partial (-r)=-\partial (r). A ''differential ring'' is a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
R equipped with one or more derivations that commute pairwise; that is, \partial_1(\partial_2 (r))=\partial_2(\partial_1 (r)) for every pair of derivations and every r\in R. 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 \Q of the rational numbers. Equivalently, this is a differential algebra over \Q, since \Q can be considered as a differential field on which every derivation is the
zero function 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
. The constants of a differential ring are the elements r such that \partial r=0 for every derivation \partial. The constants of a differential ring form a
subring In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...
and the constants of a differentiable field form a subfield. This meaning of "constant" generalizes the concept of a
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. Basic properties As a real-valued function of a real-valued argument, a constant function has the general form or just For example, ...
, and must not be confused with the common meaning of a constant.


Basic formulas

In the following identities, \delta is a derivation of a differential ring R. * If r\in R and c is a constant in R (that is, \delta c=0), then \delta (c r)= c \delta (r). * If r\in R and u is a
unit Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
in R, then \delta \left( \frac \right)= \frac * If n is a nonnegative integer and r\in R then \delta (r^)= n r^ \delta (r) * If u_1, \ldots, u_n are units in R, and e_1, \ldots, e_n are integers, one has the
logarithmic derivative In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function is defined by the formula \frac where is the derivative of . Intuitively, this is the infinitesimal relative change in ; that is, the in ...
identity:
\frac = e_ \frac + \dots + e_ \frac.


Higher-order derivations

A derivation operator or higher-order derivation is the
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...
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 \delta_1^ \circ \cdots \circ \delta_n^, where \delta_1, \ldots, \delta_n are the derivations under consideration, e_1, \ldots, e_n are nonnegative integers, and the exponent of a derivation denotes the number of times this derivation is composed in the operator. The sum o=e_1+ \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 Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
, 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, \delta_1^ \circ \cdots \circ \delta_n^(x). A proper derivative is a derivative of positive order.


Differential ideals

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, \partial x\in I, for every derivation \partial and every x\in I. A differential ideal is said to be proper if it is not the whole ring. For avoiding confusion, an ideal that is not a differential ideal is sometimes called an ''algebraic ideal''. The radical of a differential ideal is the same as its
radical Radical (from Latin: ', root) may refer to: Politics and ideology Politics *Classical radicalism, the Radical Movement that began in late 18th century Britain and spread to continental Europe and Latin America in the 19th century *Radical politics ...
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. A prime differential ideal is a differential ideal that is
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
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. 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. The algebraic ideal generated by S is the set of finite linear combinations of elements of S, and is commonly denoted as (S) or \langle S \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 When S is finite, /math> is generally not finitely generated as an algebraic ideal. The radical differential ideal generated by S is commonly denoted as \. 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 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 fraction In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmetic fractions. A ration ...
s K(X)=K(x_1,\ldots ,x_n) (fractions of multivariate polynomials), equipped with derivations \partial_i such that \partial_i x_i=1 and \partial_i x_j=0 if i\neq j (the usual partial derivatives). For defining the ring K \= K \ of differential polynomials over K with indeterminates in Y=\ with derivations \partial_1, \ldots, \partial_n, one introduces an infinity of new indeterminates of the form \Delta y_i, where \Delta is any derivation operator of order higher than . With this notation, K \ 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\=K\left , \partial y, \partial^2 y, \partial^3 y, \ldots\right Even when n=1, a ring of differential polynomials is not
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
. This makes the theory of this generalization of polynomial rings difficult. However, two facts allow such a generalization. Firstly, a finite number of differential polynomials involves together a finite number of indeterminates. Its follows that every property of polynomials that involves a finite number of polynomials remains true for differential polynomials. In particular,
greatest common divisor In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest co ...
s exist, and a ring of differential polynomials is a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
. 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 In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings. These conditions p ...
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), that satisfies the ascending chain condition on radical differential ideals, then the ring of differential polynomials R\ satisfies the same property (one passes from the univariate to the multivariate case by applying the theorem iteratively). This Noetherian property implies that, in a ring of differential polynomials, every radical differential ideal is finitely generated as a radical differential ideal; this means that there exists a finite set of differential polynomials such that is the smallest radical differential ideal containing . 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.


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 In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over th ...
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 In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( re ...
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 In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all ( monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., * If u \leq v an ...
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(\theta_\mu) \ge \operatorname(\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_\text, determines the derivative's rank. : \eta(\delta_1^ \circ \cdots \circ \delta_n^(y_j))= (j, e_1, \ldots, e_n) . : \eta(\theta_\mu y_j) \ge_\text \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_ \cdot u_p^ + \cdots +a_1 \cdot u_p+ a_0 . * Leader or leading derivative is the polynomial's highest ranked derivative: u_p. *
Coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s a_d, \ldots, a_0 do not contain the leading derivative u_p. * Degree of polynomial is the leading derivative's greatest exponent: \deg_(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. Separant set is S_A= \ , initial set is I_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 \ \setminus \mathcal, 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_ 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 A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional ...
meaning each polynomial element has a distinct leading derivative. Ritt's reduction algorithm identifies integers i_, s_ and transforms a differential polynomial f using pseudodivision to a lower or equally ranked remainder polynomial f_ 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_\text \equiv \prod_ I_^ \cdot S_^ \cdot f, \pmod \text i_, s_ \in \mathbb.


Ranking polynomial sets

Set A is a differential chain if the rank of the leading derivatives is u_ < \dots < u_ and \forall i, \ A_ is reduced with respect to A_ 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 < \cdots < A_m \in A and B_1 < \cdots < B_n \in B and i,j,k \in \mathbb. * \text A < \text B if there is a k \le \operatorname(m,n) such that A_i = B_i for 1 \le i < k and A_k < B_k . * \operatorname A < \operatorname B if n < m and A_i = B_i for 1 \le i \le n . * \operatorname A = \operatorname 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. The delta polynomial applies to polynomial pair p,q whose leaders share a common derivative, \theta_ u_= \theta_ u_. The least common derivative operator for the polynomial pair's leading derivatives is \theta_, and the delta polynomial is: : \operatorname(p,q)= S_ \cdot \frac - S_ \cdot \frac 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_ \supseteq H_A with set H_\Omega reduced with respect to the equation set. Regular differential ideal \mathcal_\text and regular algebraic ideal \mathcal_\text 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_\text= H_\Omega^\infty. * Regular algebraic ideal: \mathcal_\text=(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 In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansions of solutions to the differential equations.


Examples


Differential fields

Example 1: (\operatorname(\operatorname(y), \partial_ )) is the differential
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are ''poles'' of the function. ...
field with a single standard derivation. Example 2: (\mathbb \, p(y)\cdot \partial_ ) is a differential field with a linear differential operator as the derivation, for any polynomial p(y) .


Derivation

Define E^(p(y))=p(y+a) as
shift operator In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function to its translation . In time series analysis, the shift operator is called the '' lag opera ...
E^ for polynomial p(y). A shift-invariant operator T commutes with the shift operator: E^ \circ T=T \circ E^. The Pincherle derivative, a derivation of shift-invariant operator T, is T^ = T \circ y - y \circ T .


Constants

Ring of integers is (\mathbb. \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, \ \delta(m)=0 . Field of rational numbers is (\mathbb. \delta), and every rational number is a constant. * Every rational number is a quotient of integers. *: \forall r \in \mathbb, \ \exists \ a \in \mathbb, \ b \in \mathbb/ \, \ r=\frac * Apply the derivation formula for quotients recognizing that derivations of integers are zero: *: \delta (r)= \delta \left ( \frac \right ) = \frac=0 .


Differential subring

Constants form the subring of constants (\mathbb, \partial_) \subset (\mathbb \, \partial_) .


Differential ideal

Element \exp(y) simply generates differential ideal exp(y) in the differential ring (\mathbb \, \partial_) .


Algebra over a differential ring

Any ring with identity is a \operatornamealgebra. Thus a differential ring is a \operatornamealgebra. If ring \mathcal is a subring of the center of unital ring \mathcal, then \mathcal is an \operatornamealgebra. 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 \, \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,\ \gcd(p, \partial_y(p))=1 : q(z)=1+z^2, \ \partial_y(q)=2 \cdot z \cdot (1+z^2),\ \gcd(q, \partial_(q))=q


Polynomials


Ranking

Ring (\mathbb \, \delta) has derivatives \delta(y_)=y_^ and \delta(y_)=y_^ * Map each derivative to an integer tuple: \eta( \delta^(y_) )=(i_, i_). * Rank derivatives and integer tuples: y_^ \ (2,2) > y_^ \ (2,1) > y_ \ (2,0) > y_^ \ (1,2) > y_^ \ (1,1) > y_ \ (1,0) .


Leading derivative and initial

The leading derivatives, and initials are: : p= \cdot ()^ + 3 \cdot y_^ \cdot + (y_^)^ : q= \cdot + y_ \cdot y_^ + (y_^)^ : r= \cdot ()^ + y_^ \cdot + 2 \cdot y_


Separants

: S_= 2 \cdot (y_+ y_^) \cdot y_^ + 3 \cdot y_^. : S_= y_+ 3 \cdot y_^ : S_= 2 \cdot (y_+3) \cdot y_^ + y_^


Autoreduced sets

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


Applications


Symbolic integration

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


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 Chaos or CHAOS may refer to: Science, technology, and astronomy * '' Chaos: Making a New Science'', a 1987 book by James Gleick * Chaos (company), a Bulgarian rendering and simulation software company * ''Chaos'' (genus), a genus of amoebae * ...
, 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. Researchers have applied differential elimination to understanding
cellular biology Cell biology (also cellular biology or cytology) is a branch of biology that studies the Anatomy, structure, Physiology, function, and behavior of cell (biology), cells. All living organisms are made of cells. A cell is the basic unit of life th ...
, compartmental biochemical models,
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
estimation and quasi-steady state approximation (QSSA) for biochemical reactions. Using differential Gröbner bases, researchers have investigated non-classical
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
properties of non-linear differential equations. Other applications include control theory, model theory, and algebraic geometry. Differential algebra also applies to differential-difference equations.


Algebras with derivations


Differential graded vector space

A \operatorname
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
V_ is a collection of vector spaces V_ with integer degree , v, =m for v\in V_. A
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
can represent this graded vector space: : V_ = \bigoplus_ V_ A differential graded vector space or
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...
, is a graded vector space V_ with a differential map or boundary map d_: V_ \to V_ with d_ \circ d_ = 0 . A
cochain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel ...
is a graded vector space V^ with a differential map or coboundary map d_: V_ \to V_ with d_ \circ d_ = 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)^ \cdot a \cdot d(b) with , a, the degree of vector a.


Lie algebra

A
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
is a finite-dimensional real or complex vector space \mathcal with a bilinear bracket operator \mathcal \times \mathcal \to \mathcal with Skew symmetry and the
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
property. * Skew symmetry: ,Y - ,X/math> * Jacobi identity property: ,[Y,Z+[Y,[Z,X">,Z.html" ;"title=",[Y,Z">,[Y,Z+[Y,[Z,X + [Z,[X,Y">,Z">,[Y,Z<_a>+[Y,[Z,X.html" ;"title=",Z.html" ;"title=",[Y,Z">,[Y,Z+[Y,[Z,X">,Z.html" ;"title=",[Y,Z">,[Y,Z+[Y,[Z,X + [Z,[X,Y=0 for all X, Y, Z \in \mathcal. The adjoint operator, \operatorname(Y)= ,X/math> is a Commutator">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_([Y,Z]) = [\operatorname_(Y),Z] + [Y,\operatorname_(Z)] The universal enveloping algebra U(\mathcal) of Lie algebra \mathcal is a maximal associative algebra with identity, generated by Lie algebra elements \mathcal 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) : X \cdot Y - Y \cdot X = ,Y/math> * Leibniz product rule: \operatorname_( Y \cdot Z)=\operatorname_(Y) \cdot Z + Y \cdot \operatorname_(Z) for all X,Y,Z \in U(\mathcal) .


Weyl algebra

The
Weyl algebra In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. ...
is an algebra A_(K) over a ring K _, q_, \dots, p_, q_/math> with a specific noncommutative product: : p_ \cdot q_ - q_ \cdot p_=1, \ : \ i \in \ . All other indeterminate products are commutative for i,j \in \: : p_ \cdot q_ - q_ \cdot p_=0 \text i \ne j, \ p_ \cdot p_ - p_ \cdot p_=0, \ q_ \cdot q_ - q_ \cdot q_=0 . A Weyl algebra can represent the derivations for a commutative ring's polynomials f \in K _, \ldots, y_/math>. The Weyl algebra's elements are
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
s, the elements p_, \ldots, p_ function as standard derivations, and map compositions generate linear differential operators.
D-module In mathematics, a ''D''-module is a module (mathematics), module over a ring (mathematics), ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. S ...
is a related approach for understanding differential operators. The endomorphisms are: : q_ (y_)= y_ \cdot y_, \ q_(c)= c \cdot y_ \text c \in K, \ p_(y_)=1, \ p_(y_)=0 \text j \ne k, \ p_(c)= 0 \text c \in K


Pseudodifferential operator ring

The associative, possibly noncommutative ring A has derivation d: A \to A . The
pseudo-differential operator In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in m ...
ring
A((\partial^)) is a left \operatorname containing ring elements L: : a_i \in A, \ i,i_ \in \mathbb, \ , i_, > 0 \ : \ L= \sum_^n a_i \cdot \partial^i The derivative operator is d(a) = \partial \circ a - a \circ \partial . The
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
is \Bigl( \Bigr). Pseudo-differential operator multiplication is: : \sum_^n a_i \cdot \partial^i \cdot \sum_^m b_ \cdot \partial^j = \sum_ \Bigl( \Bigr) \cdot a_i \cdot d^k(b_j) \cdot \partial^


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. 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.


See also

* * * * * * * * * * * * * Kolchin's problems


Citations


References

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * {{refend


External links


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