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.
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.
, one has the identity function, which is generally considered as the unique derivation operator of order zero.
This makes the theory of this generalization of polynomial rings difficult.
In particular, greatest common divisors exist, and a ring of differential polynomials is a unique factorization domain.
contains the field of rational numbers, the rings of differential polynomials over
satisfy the ascending chain condition on radical differential ideals.
satisfies the same property (one passes from the univariate to the multivariate case by applying the theorem iteratively).
Types of ranking include:[27] In this example, the integer tuple identifies the differential indeterminate and derivative's multi-index, and lexicographic monomial order,
An autoreduced set is triangular meaning each polynomial element has a distinct leading derivative.
using pseudodivision to a lower or equally ranked remainder polynomial
The algorithm's first step partially reduces the input polynomial and the algorithm's second step fully reduces the polynomial.
These regular differential radical ideals, represented by characteristic sets, are not necessarily prime ideals and the representation is not necessarily minimal.
is a member of an ideal generated from a set of differential polynomials
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.
[40] The Rosenfeld–Gröbner algorithm facilitates creating Taylor series expansions of solutions to the differential equations.
is the differential meromorphic function field with a single standard derivation.
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.
Methods are available to facilitate the numerical integration of a differential-algebraic system of equations.
They were successful in most cases, and this facilitated developing approximate solutions, efficiently evaluating chaos, and constructing Lyapunov functions.
[48] Researchers have applied differential elimination to understanding cellular biology, compartmental biochemical models, parameter estimation and quasi-steady state approximation (QSSA) for biochemical reactions.
[57] A Lie algebra is a finite-dimensional real or complex vector space
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.
The adjoint operator is a derivation following the Leibniz product rule.
: A Weyl algebra can represent the derivations for a commutative ring's polynomials
function as standard derivations, and map compositions generate linear differential operators.
D-module is a related approach for understanding differential operators.
, a long gap chain of irreducible differential algebraic subvarieties occurs from