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.[1][2][3]

A natural example of a differential field is the field of rational functions in one variable over the complex numbers, 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.



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.[4] 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.[5][6][2] Ellis Kolchin, Ritt's student, advanced this field and published Differential Algebra And Algebraic Groups.[1]

Differential rings




A derivation   on a ring   is a function   such that

  (Leibniz product rule),

for every   and   in  

A derivation is linear over the integers since these identities imply   and  

A differential ring is a commutative ring   equipped with one or more derivations that commute pairwise; that is,

for every pair of derivations and every  [7] 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   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.

The constants of a differential ring are the elements   such that   for every derivation   The constants of a differential ring form a subring and the constants of a differentiable field form a subfield.[8] 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,   is a derivation of a differential ring  [9]

  • If   and   is a constant in   (that is,  ), then
  • If   and   is a unit in   then
  • If   is a nonnegative integer and   then
  • If   are units in   and   are integers, one has the logarithmic derivative identity:

Higher-order derivations


A derivation operator or higher-order derivation[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

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, 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.[7]

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 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.[10] 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.[11] 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.[11][12]

The algebraic ideal generated by   is the set of the 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,   is generally not finitely generated as an algebraic ideal.

The radical differential ideal generated by   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   is a formalization of the concept of differential equation such that the known functions appearing in the equation belong to   and the indeterminates are symbols for the unknown functions.

So, let   be a differential field, which is typically (but not necessarily) a field of rational fractions   (fractions of multivariate polynomials), equipped with derivations   such that   and   if   (the usual partial derivatives).

For defining the ring   of differential polynomials over   with indeterminates in   with derivations   one introduces an infinity of new indeterminates of the form   where   is any derivation operator of order higher than 1. With this notation,   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   one has


Even when   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 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 divisors exist, and a ring of differential polynomials is a unique factorization domain.

The second fact is that, if the field   contains the field of rational numbers, the rings of differential polynomials over   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   is a Ritt Algebra (that, is a differential ring containing the field of rational numbers),[13] that satisfies the ascending chain condition on radical differential ideals, then the ring of differential polynomials   satisfies the same property (one passes from the univariate to the multivariate case by applying the theorem iteratively).[14][15]

This Noetherian property implies that, in a ring of differential polynomials, every radical differential ideal I is finitely generated as a radical differential ideal; this means that there exists a finite set S of differential polynomials such that I is the smallest radical differential idesl containing S.[16] 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.[17]

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.[1][18][19][20][21][22][23]

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:[24][25][26]


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:[27]

  • Orderly ranking:  
  • Elimination ranking:  

In this example, the integer tuple identifies the differential indeterminate and derivative's multi-index, and lexicographic monomial order,  , determines the derivative's rank.[28]


Leading derivative, initial and separant


This is the standard polynomial form:  .[24][28]

  • Leader or leading derivative is the polynomial's highest ranked derivative:  .
  • Coefficients   do not contain the leading derivative  .
  • Degree of polynomial is the leading derivative's greatest exponent:  .
  • Initial is the coefficient:  .
  • Rank is the leading derivative raised to the polynomial's degree:  .
  • Separant is the derivative:  .

Separant set is  , initial set is   and combined set is  .[29]



Partially reduced (partial normal form) polynomial   with respect to polynomial   indicates these polynomials are non-ground field elements,  , and   contains no proper derivative of  .[30][31][29]

Partially reduced polynomial   with respect to polynomial   becomes reduced (normal form) polynomial   with respect to   if the degree of   in   is less than the degree of   in  .[30][31][29]

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.[32][30]

Ritt's reduction algorithm identifies integers   and transforms a differential polynomial   using pseudodivision to a lower or equally ranked remainder polynomial   that is reduced with respect to the autoreduced polynomial set  . 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:[30]


Ranking polynomial sets


Set   is a differential chain if the rank of the leading derivatives is   and   is reduced with respect to  [33]

Autoreduced sets   and   each contain ranked polynomial elements. This procedure ranks two autoreduced sets by comparing pairs of identically indexed polynomials from both autoreduced sets.[34]

  •   and   and  .
  •   if there is a   such that   for   and  .
  •   if   and   for  .
  •   if   and   for  .

Polynomial sets


A characteristic set   is the lowest ranked autoreduced subset among all the ideal's autoreduced subsets whose subset polynomial separants are non-members of the ideal  .[35]

The delta polynomial applies to polynomial pair   whose leaders share a common derivative,  . The least common derivative operator for the polynomial pair's leading derivatives is  , and the delta polynomial is:[36][37]


A coherent set is a polynomial set that reduces its delta polynomial pairs to zero.[36][37]

Regular system and regular ideal


A regular system   contains a autoreduced and coherent set of differential equations   and a inequation set   with set   reduced with respect to the equation set.[37]

Regular differential ideal   and regular algebraic ideal   are saturation ideals that arise from a regular system.[37] Lazard's lemma states that the regular differential and regular algebraic ideals are radical ideals.[38]

  • Regular differential ideal:  
  • Regular algebraic ideal:  

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.[39]

The membership problem is to determine if a differential polynomial   is a member of an ideal generated from a set of differential polynomials  . 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.[40]

The Rosenfeld–Gröbner algorithm facilitates creating Taylor series expansions of solutions to the differential equations.[41]



Differential fields


Example 1:   is the differential meromorphic function field with a single standard derivation.

Example 2:   is a differential field with a linear differential operator as the derivation.



Define   as shift operator   for polynomial  .

A shift-invariant operator   commutes with the shift operator:  .

The Pincherle derivative, a derivation of shift-invariant operator  , is  .[42]



Ring of integers is  , and every integer is a constant.

  • The derivation of 1 is zero.  .
  • Also,  .
  • By induction,  .

Field of rational numbers is  , and every rational number is a constant.

  • Every rational number is a quotient of integers.
  • Apply the derivation formula for quotients recognizing that derivations of integers are zero:

Differential subring


Constants form the subring of constants  .[43]

Differential ideal


Element   simply generates differential ideal   in the differential ring  .[44]

Algebra over a differential ring


Any ring with identity is a  algebra.[45] Thus a differential ring is a  algebra.

If ring   is a subring of the center of unital ring  , then   is an  algebra.[45] Thus, a differential ring is an algebra over its differential subring. This is the natural structure of an algebra over its subring.[30]

Special and normal polynomials


Ring   has irreducible polynomials,   (normal, squarefree) and   (special, ideal generator).






Ring   has derivatives   and  

  • Map each derivative to an integer tuple:  .
  • Rank derivatives and integer tuples:  .

Leading derivative and initial


The leading derivatives, and initials are:




Autoreduced sets

  • Autoreduced sets are   and  . Each set is triangular with a distinct polynomial leading derivative.
  • The non-autoreduced set   contains only partially reduced   with respect to  ; 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.[46]

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.[47]

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 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.[49][50] Using differential Gröbner bases, researchers have investigated non-classical symmetry properties of non-linear differential equations.[51] Other applications include control theory, model theory, and algebraic geometry.[52][16][53] Differential algebra also applies to differential-difference equations.[54]

Algebras with derivations


Differential graded vector space


A   vector space   is a collection of vector spaces   with integer degree   for  . A direct sum can represent this graded vector space:[55]


A differential graded vector space or chain complex, is a graded vector space   with a differential map or boundary map   with   .[56]

A cochain complex is a graded vector space   with a differential map or coboundary map   with  .[56]

Differential graded algebra


A differential graded algebra is a graded algebra   with a linear derivation   with   that follows the graded Leibniz product rule.[57]

  • Graded Leibniz product rule:   with   the degree of vector  .

Lie algebra


A Lie algebra is a finite-dimensional real or complex vector space   with a bilinear bracket operator   with Skew symmetry and the Jacobi identity property.[58]

  • Skew symmetry:  
  • Jacobi identity property:  

for all  .

The adjoint operator,   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  .[59][60]


The universal enveloping algebra   of Lie algebra   is a maximal associative algebra with identity, generated by Lie algebra elements   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.[61]

  • Product in   :  
  • Leibniz product rule:  

for all  .

Weyl algebra


The Weyl algebra is an algebra   over a ring   with a specific noncommutative product: [62]


All other indeterminate products are commutative for  :


A Weyl algebra can represent the derivations for a commutative ring's polynomials  . The Weyl algebra's elements are endomorphisms, the elements   function as standard derivations, and map compositions generate linear differential operators. D-module is a related approach for understanding differential operators. The endomorphisms are:[62]


Pseudodifferential operator ring


The associative, possibly noncommutative ring   has derivation  .[63]

The pseudo-differential operator ring   is a left   containing ring elements  :[63][64][65]


The derivative operator is  .[63]

The binomial coefficient is  .

Pseudo-differential operator multiplication is:[63]


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.[66]

The Kolchin catenary conjecture states given a   dimensional irreducible differential algebraic variety   and an arbitrary point  , a long gap chain of irreducible differential algebraic subvarieties occurs from   to V.[67]

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.[68]

See also



  1. ^ a b c Kolchin 1973
  2. ^ a b Ritt 1950
  3. ^ Kaplansky 1976
  4. ^ Ritt 1932, pp. iii–iv
  5. ^ Ritt 1930
  6. ^ Ritt 1932
  7. ^ a b Kolchin 1973, pp. 58–59
  8. ^ Kolchin 1973, pp. 58–60
  9. ^ Bronstein 2005, p. 76
  10. ^ Sit 2002, pp. 3–4
  11. ^ a b Kolchin 1973, pp. 61–62
  12. ^ Buium 1994, p. 21
  13. ^ Kaplansky 1976, p. 12
  14. ^ Kaplansky 1976, pp. 45, 48, 56–57
  15. ^ Kolchin 1973, pp. 126–129
  16. ^ a b Marker 2000
  17. ^ Hubert 2002, p. 8
  18. ^ Li & Yuan 2019
  19. ^ Boulier et al. 1995
  20. ^ Mansfield 1991
  21. ^ Ferro 2005
  22. ^ Chardin 1991
  23. ^ Wu 2005b
  24. ^ a b Kolchin 1973, pp. 75–76
  25. ^ Gao et al. 2009, p. 1141
  26. ^ Hubert 2002, p. 10
  27. ^ Ferro & Gerdt 2003, p. 83
  28. ^ a b Wu 2005a, p. 4
  29. ^ a b c Boulier et al. 1995, p. 159
  30. ^ a b c d e Kolchin 1973, p. 75
  31. ^ a b Ferro & Gerdt 2003, p. 84
  32. ^ Sit 2002, p. 6
  33. ^ Li & Yuan 2019, p. 294
  34. ^ Kolchin 1973, p. 81
  35. ^ Kolchin 1973, p. 82
  36. ^ a b Kolchin 1973, p. 136
  37. ^ a b c d Boulier et al. 1995, p. 160
  38. ^ Morrison 1999
  39. ^ Boulier et al. 1995, p. 158
  40. ^ Boulier et al. 1995, p. 164
  41. ^ Boulier et al. 2009b
  42. ^ Rota, Kahaner & Odlyzko 1973, p. 694
  43. ^ Kolchin 1973, p. 60
  44. ^ Sit 2002, p. 4
  45. ^ a b Dummit & Foote 2004, p. 343
  46. ^ Bronstein 2005, pp. 41, 51, 53, 102, 299, 309
  47. ^ Hubert 2002, pp. 41–47
  48. ^ Harrington & VanGorder 2017
  49. ^ Boulier 2007
  50. ^ Boulier & Lemaire 2009a
  51. ^ Clarkson & Mansfield 1994
  52. ^ Diop 1992
  53. ^ Buium 1994
  54. ^ Gao et al. 2009
  55. ^ Keller 2019, p. 48
  56. ^ a b Keller 2019, pp. 50–51
  57. ^ Keller 2019, pp. 58–59
  58. ^ Hall 2015, p. 49
  59. ^ Hall 2015, p. 51
  60. ^ Jacobson 1979, p. 9
  61. ^ Hall 2015, p. 247
  62. ^ a b Lam 1991, pp. 7–8
  63. ^ a b c d Parshin 1999, p. 268
  64. ^ Dummit & Foote 2004, p. 337
  65. ^ Taylor 1991
  66. ^ Golubitsky, Kondratieva & Ovchinnikov 2009
  67. ^ Freitag, Sánchez & Simmons 2016
  68. ^ Lando 1970


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  • Taylor, Michael E. (1991). Pseudodifferential operators and nonlinear PDE. Boston: Birkhäuser. ISBN 978-0-8176-3595-4.
  • Wu, Wen-tsün (2005a). "On "Good" Bases of Algebraic-Differential Ideals". Differential equations with symbolic computation. Birkhäuser. pp. 343–350. doi:10.1007/3-7643-7429-2_19. ISBN 978-3-7643-7429-7.
  • Wu, Wen-tsün (2005b). "On the Construction of Groebner Basis of a Polynomial Ideal Based on Riquier–Janet Theory". Differential equations with symbolic computation. Trends in Mathematics. Birkhäuser. pp. 351–368. doi:10.1007/3-7643-7429-2_20. ISBN 978-3-7643-7429-7.
  • Zharinov, V. V. (December 2021). "Navier–Stokes equations, the algebraic aspect" (PDF). Theoretical and Mathematical Physics. 209 (3): 1657–1672. arXiv:2110.01504. Bibcode:2021TMP...209.1657Z. doi:10.1134/S0040577921120011. S2CID 238259977.
  • David Marker's home page has several online surveys discussing differential fields.