In
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,
where the derivation is differentiation with respect to
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 ...
on a ring
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 ...
such that
and
:
(
Leibniz product rule),
for every
and
in
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
and
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 ...
equipped with one or more derivations that commute pairwise; that is,
for every pair of derivations and every
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''
over a differential field
is a differential ring that contains
as a subring such that the restriction to
of the derivations of
equal the derivations of
(A more general definition is given below, which covers the case where
is not a field, and is essentially equivalent when
is a field.)
A ''Witt algebra'' is a differential ring that contains the field
of the rational numbers. Equivalently, this is a differential algebra over
since
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
such that
for every derivation
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,
is a derivation of a differential ring
* If
and
is a constant in
(that is,
), then
* If
and
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
then
* If
is a nonnegative integer and
then
* If
are units in
and
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:
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
where
are the derivations under consideration,
are nonnegative integers, and the exponent of a derivation denotes the number of times this derivation is composed in the operator.
The sum
is called the ''order'' of derivation. If
the derivation operator is one of the original derivations. If
, 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
of a differential ring is the application of a derivation operator to
that is, with the above notation,
A
proper derivative is a derivative of positive order.
Differential ideals
A
differential ideal of a differential ring
is an
ideal of the ring
that is
closed (stable) under the derivations of the ring; that is,
for every derivation
and every
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
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
is the set of finite linear combinations of elements of
and is commonly denoted as
or
The differential ideal generated by
is the set of the finite linear combinations of elements of
and of the derivatives of any order of these elements; it is commonly denoted as
When
is finite,