Differential algebra

This article is about algebraic study of differential equations. For the concept in homological algebra, see Differential graded algebra.

In mathematics, differential algebra is, broadly speaking, the study of rings with extra operations includin. This for example includes difference algebra and wittferential algebra which is about p-derivations.

Classical, differential algebra, which this article will focus on, is 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. Differential algebra is to is to Differential Algebraic Geometry (the study of the geometry of algebraic differential equations) as Commutative Algebra is to Algebraic Geometry. The theory has close connections to the Theory of Holomorphic Foliations in the same way that Algebraic Geometry is closely related to Complex Geometry (the study of complex manifolds).

Let $\Delta =\lbrace \partial _{1},\ldots ,\partial _{m}\rbrace$ be a collection of commuting formal derivations on a ring $R$. That is, each $\partial \in \Delta$ satisfies a sum rule $\partial (f+g)=\partial (f)+\partial (g)$, product rule $\partial (fg)=\partial (f)g+f\partial (g)$, and $\partial (0)=0,\partial (1)=0$ .

Often writers restrict to the case of a single derivation when $\Delta =\lbrace \partial \rbrace$ and when these two cases need to be distinguished are referred to as "ordinary differential algebra" or "partial differential algebra" corresponding to the terminology of ordinary and partial differential equations.

A natural example of a differential field is the field of rational functions in one variable over the complex numbers, $\mathbb {C} (t),$ 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

Many theorem from Differential algebra are quite old and go as far back as to Newton, Euler, Lagrange. Some of its modern problems date back to Poincare and Jacobi. For example the problem of when integrals can actually be integrated and why $\int e^{x^{2}}dx$ has no solution in "elementary functions" was investigated by Liouville. The galois theory for linear differential equations is closely tied to the monodromy group where fundamental matrices change after winding around a loop. In the case of irregular singularities this leads the the study of Stokes Phenomena (see Stokes Parameters) which has deep connections to the theory of D-Modules developed by Bernard Malgrange and others.

Its modern form, which mirrors algebraic geometry, started with Joseph Ritt. 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 ${\textstyle \partial }$ on a ring ${\textstyle R}$ is a function $\partial :R\to R\,$ 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 over the integers since these identities imply $\partial (0)=\partial (1)=0$ and $\partial (-r)=-\partial (r).$

A differential ring is a commutative ring $R$ 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 differentiable 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 Ritt algebra is a differential ring that contains the field $\mathbb {Q}$ of the rational numbers. Equivalently, this is a differential algebra over $\mathbb {Q} ,$ since $\mathbb {Q}$ can be considered as a differential field on which every derivation is the zero function. This isn't a central concept but sometimes appears.)

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 and the constants of a differentiable field form a subfield. This meaning of "constant" generalizes the concept of a constant function, and must not be confused with the common meaning of a constant.

Advanced Setup

Ring with general operations (e.g. difference rings or p-derivations) can be encoded with Bialgebras (usually called Birings in this literature). Tall, David O., and Gavin C. Wraith. "Representable functors and operations on rings." Proceedings of the London Mathematical Society 3.4 (1970): 619-643.

The classification of strongly minimal differential algebraic varieties. This is related to the Model Theory of Differential equations and Zariski Geometries.

Ferra Carro's Problem on the existence of differential Groebner bases.

The p-Curvature Conjecture of Katz and Grothendieck.

Higher order versions of the Painleve equations with no movable singularities.

See Kolchin's Problems.

See also

• Arithmetic derivative – Function defined on integers in number theory
• Difference algebra
• Differential algebraic geometry
• Differential calculus over commutative algebras – part of commutative algebraPages displaying wikidata descriptions as a fallback
• Differential Galois theory – Study of Galois symmetry groups of differential fields
• Differentially closed field
• Differential graded algebra – Algebraic structure in homological algebra
• D-module – Module over a sheaf of differential operators
• Hardy field – Mathematical concept
• Kähler differential – Differential form in commutative algebra
• Liouville's theorem (differential algebra) – Says when antiderivatives of elementary functions can be expressed as elementary functions
• Picard–Vessiot theory – Study of differential field extensions induced by linear differential equations

References

• Boulier, François; Lazard, Daniel; Ollivier, François; Petitot, Michel (1995). "Representation for the radical of a finitely generated differential ideal". Proceedings of the 1995 international symposium on Symbolic and algebraic computation – ISSAC '95 (PDF). pp. 158–166. doi:10.1145/220346.220367. ISBN 0897916999. S2CID 11059042.
• Boulier, François (31 December 2007). "Differential Elimination and Biological Modelling". Gröbner Bases in Symbolic Analysis. 2: 109–138. doi:10.1515/9783110922752.109. ISBN 978-3-11-019323-7. S2CID 61916692.
• Boulier, François; Lemaire, François (2009a). "Differential algebra and QSSA methods in biochemistry". IFAC Proceedings Volumes. 42 (10): 33–38. doi:10.3182/20090706-3-FR-2004.00004.
• Boulier, François; Lazard, Daniel; Ollivier, François; Petitot, Michel (April 2009b). "Computing representations for radicals of finitely generated differential ideals". Applicable Algebra in Engineering, Communication and Computing. 20 (1): 73–121. doi:10.1007/s00200-009-0091-7. S2CID 5482290.
• Bronstein, Manuel (2005). Symbolic integration I : transcendental functions. Algorithms and Computation in Mathematics. Vol. 1 (2nd ed.). Berlin: Springer. doi:10.1007/b138171. ISBN 3-540-21493-3.
• Buium, Alexandru (1994). Differential algebra and diophantine geometry. Hermann. ISBN 978-2-7056-6226-4.
• Chardin, Marc (1991). "Differential resultants and subresultants". In Budach, L. (ed.). Fundamentals of Computation Theory. FCT 1991. Lecture Notes in Computer Science. Vol. 529. Berlin, Heidelberg: Springer. pp. 180–189. doi:10.1007/3-540-54458-5_62. ISBN 978-3-540-38391-8.
• Clarkson, Peter A.; Mansfield, Elizabeth L. (January 1994). "Symmetry reductions and exact solutions of a class of nonlinear heat equations". Physica D: Nonlinear Phenomena. 70 (3): 250–288. arXiv:solv-int/9306002. Bibcode:1994PhyD...70..250C. doi:10.1016/0167-2789(94)90017-5. S2CID 16858637.
• Crespo, Teresa; Hajto, Zbigniew (2011). Algebraic groups and differential Galois theory. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-5318-4.
• Diop, Sette (May 1992). "Differential-algebraic decision methods and some applications to system theory" (PDF). Theoretical Computer Science. 98 (1): 137–161. doi:10.1016/0304-3975(92)90384-R.
• Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons. ISBN 0-471-43334-9.
• Ferro, Giuseppa Carrá; Gerdt, V. P. (2003). "Improved Kolchin–Ritt Algorithm". Programming and Computer Software. 29 (2): 83–87. doi:10.1023/A:1022996615890. S2CID 26280002.
• Ferro, Giuseppa Carrá (2005). "Generalized Differential Resultant Systems of Algebraic ODEs and Differential Elimination Theory". Differential equations with symbolic computation. Trends in Mathematics. Birkhäuser. pp. 343–350. doi:10.1007/3-7643-7429-2_18. ISBN 978-3-7643-7429-7.
• Freitag, James; Sánchez, Omar León; Simmons, William (2 June 2016). "On Linear Dependence Over Complete Differential Algebraic Varieties". Communications in Algebra. 44 (6): 2645–2669. arXiv:1401.6211. doi:10.1080/00927872.2015.1057828. S2CID 56218725.
• Gao, X. S.; Van der Hoeven, J.; Yuan, C. M.; Zhang, G. L. (1 September 2009). "Characteristic set method for differential–difference polynomial systems". Journal of Symbolic Computation. 44 (9): 1137–1163. doi:10.1016/j.jsc.2008.02.010.
• Golubitsky, O. D.; Kondratieva, M. V.; Ovchinnikov, A. I. (2009). "On the generalized Ritt problem as a computational problem". Journal of Mathematical Sciences. 163 (5): 515–522. arXiv:0809.1128. doi:10.1007/s10958-009-9689-3. S2CID 17503904.
• Hall, Brian C. (2015). Lie groups, Lie algebras, and representations: an elementary introduction (Second ed.). Cham: Springer. ISBN 978-3-319-13467-3.
• Harrington, Heather A.; VanGorder, Robert A. (2017). "Reduction of dimension for nonlinear dynamical systems". Nonlinear Dynamics. 88 (1): 715–734. doi:10.1007/s11071-016-3272-5. PMC 7089670. PMID 32226227. S2CID 254893812.
• Hubert, Evelyne (2002). "Notes on Triangular Sets and Triangulation-Decomposition Algorithms II: Differential Systems". In Winkler, Franz; Langer, Ulrich (eds.). Symbolic and Numerical Scientific Computing. Second International Conference, SNSC 2001 Hagenberg, Austria, September 12-14, 2001 Revised Papers (PDF). Berlin: Springer-Verlag. pp. 40–87. ISBN 3-540-40554-2.
• Jacobson, Nathan (1979). Lie algebras. New York. ISBN 0-486-63832-4.{{cite book}}: CS1 maint: location missing publisher (link)
• Kaplansky, Irving (1976). An introduction to differential algebra (2nd ed.). Hermann. ISBN 9782705612511.
• Keller, Corina (2019). Chern-Simons theory and equivariant factorization algebras. BestMasters. Wiesbaden, Germany. doi:10.1007/978-3-658-25338-7. ISBN 978-3-658-25337-0. S2CID 128325519.{{cite book}}: CS1 maint: location missing publisher (link)
• Kolchin, Ellis (1973). Differential Algebra And Algebraic Groups. Academic Press. ISBN 978-0-08-087369-5.
• Lam, T. Y. (1991). A first course in noncommutative rings. Graduate Texts in Mathematics. Vol. 131. New York: Springer-Verlag. doi:10.1007/978-1-4419-8616-0. ISBN 0-387-97523-3.
• Lando, Barbara A. (1970). "Jacobi's bound for the order of systems of first order differential equations". Transactions of the American Mathematical Society. 152 (1): 119–135. doi:10.1090/S0002-9947-1970-0279079-1. ISSN 0002-9947.
• Li, Wei; Yuan, Chun-Ming (February 2019). "Elimination Theory in Differential and Difference Algebra". Journal of Systems Science and Complexity. 32 (1): 287–316. doi:10.1007/s11424-019-8367-x. S2CID 255158214.
• Marker, David (2000). "Model theory of differential fields". In Haskell, Deirdre; Pillay, Anand; Steinhorn, Charles (eds.). Model theory, algebra, and geometry (PDF). Vol. 39. Cambridge: Cambridge University Press. pp. 53–64. ISBN 0-521-78068-3.
• Mansfield, Elizabeth (1991). Differential Bases (PhD). University of Sydney.
• Morrison, Sally (1 October 1999). "The Differential Ideal [ P ] : M∞" (PDF). Journal of Symbolic Computation. 28 (4): 631–656. doi:10.1006/jsco.1999.0318. ISSN 0747-7171.
• Parshin, Aleksei Nikolaevich (1999). "On a ring of formal pseudo-differential operators". Proc. Steklov Math. Institute. 224: 266–280. arXiv:math/9911098. Bibcode:1999math.....11098P.
• Ritt, Joseph Fels (1930). "Manifolds of functions defined by systems of algebraic differential equations" (PDF). Transactions of the American Mathematical Society. 32 (4): 569–598. doi:10.1090/S0002-9947-1930-1501554-4. S2CID 54064812.
• Ritt, Joseph (1932). differential equations from the algebraic standpoint. Vol. 14. American Mathematical Society.
• Ritt, Joseph Fels (1950). Differential Algebra. Vol. 33. Providence, Rhode Island: American Mathematical Society Colloquium Publications. ISBN 978-0-8218-3205-9. {{cite book}}: ISBN / Date incompatibility (help)
• Rota, Gian-Carlo; Kahaner, David; Odlyzko, Andrew (1973). "On the foundations of combinatorial theory. VIII. Finite operator calculus". Journal of Mathematical Analysis and Applications. 42 (3): 684–760. doi:10.1016/0022-247X(73)90172-8.
• Sit, William Y. (2002). "The Ritt-Kolchin theory for differential polynomials". In Guo, Li; Cassidy, Phyllis J; Keigher, William F; Sit, William Y (eds.). Differential algebra and related topics: proceedings of the International Workshop, Newark Campus of Rutgers, the State University of New Jersey, 2-3 November 2000. River Edge, NJ: World Scientific. doi:10.1142/4768. ISBN 981-02-4703-6.
• Stechlinski, Peter; Patrascu, Michael; Barton, Paul I. (2018). "Nonsmooth differential-algebraic equations in chemical engineering". Computers & Chemical Engineering. 114: 52–68. doi:10.1016/j.compchemeng.2017.10.031. hdl:1721.1/122980. S2CID 49413118.
• 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.

External links

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