First presenter Co-presenter(s)
Name :  Lu Yang Name:  Zhenbing Zeng
E-mail: E-mail:  
Affiliation: East China Normal University Name:  Weinian Zhang
Department: Shanghai Key Laboratory of Trustworthy Computing  E-mail:    
City: Shanghai 200062 Name:   
State/Province:   E-mail:    
Country: P. R. China Name:   
04-07  E-mail:    
Session: 4- Elimination Theory and Applications Schedule:
Saturday, 14:00
Related website:  
Title of
An Attempt to Apply Dixon Resultant to Differential Elimination

The idea of using resultant for solving differential equations can be traced to 1930's. The first work on "differential resultant" was done by O. Ore [19]. Later, the divisibility and elimination theory of algebraic differential equations was developed by C. H. Riquier [20], M. Janet [8], J. F. Ritt [21, 22], E. R. Kolchin [12] et al. The notion of resultant of two nonlinear differential polynomials was introduced by Ritt [21] in 1932 under some hypothesis on the differential polynomials by using of pseudo-division. In 1997 G. Carra-Ferro [1] introduced a notion of differential resultant of two ordinary differential polynomials as quotient of two determinants and proved that a necessary condition for the existence of a common solution of two algebraic differential equations is that the differential resultant is equal to zero based on Macaulay's resultant (see [7, 17, 18]). Methods for using characteristic set theory to solve first order autonomous ODE and partial difference polynomial systems was propsed by X. S. Gao et al in [6]. Other generalizations of differential resultant can be found in [3], [11].

In this paper, we present our joint work on computing differential resultant via Dixon's resultant [10] for first and higher ODEs. Comparing with the classical resultant of Sylvester, the advantage of Dixon's resultant is that it can do one-step elimination of a block of unknowns from a system of polynomial equations like Macauley's. Meanwhile, there are works showing that Dixon's resultant is much faster than Macauley's for certain specific problems (cf. [13, 14, 15, 16]). So the motivation for using Dixon's resultant to solve differential resultants comes very natural. Our results in this paper show that this attempt is also worth to develop for many non-trivial problems listed in E. Kamke's book [9].

[1] G. Carra-Ferro, A resultant theory for the systems of two ordinary algebraic differential equations, Appl. Algebra Engin. Comm. Comput. 8(1997), 539-560.
[3] G. Carra-Ferro, Generalized differential resultant systems of algebraic ODEs and differential elimination theory, Trends in Mathematics: Differential Equations with Symbolic Computation, Birkhauser Verlag Basel, 2006, 327-334.
[5] Mao-Ching Foo and Eng-Wee Chionh, Corner edge cutting and Dixon-resultant quotients, J. Symbolic Computation 37(2004), no. 1, 101-119.
[6] X.S. Gao and M.B. Zhang, Decomposition of differential polynomials with constant coefficients, Proc. ISSAC (2004), 175-182.
[7] I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Discriminants Resultants and Multidimensional Determinants, Birkhauser, Berlin, 1994.
[8] M. Janet, Sur les sist'emes d''equations aux d'eriv'ees partielles, J. Math. 3(1920), 65-151.
[9] E. Kamke, Differentialgleichungen, osungsmethoden und osungen, Akad. Verlag. Geest & Portig K.-G., Leipzig, 1956.
[10] D. Kapur, T. Saxena and Lu Yang, Algebraic and geometric reasoning using Dixon resultants, Proc. ISSAC (1994), 99-107.
[11] A. Kasman and E. Previato, Commutative partial differential operators, Physica D 152- 153 (2001), 66-77.
[12] E. R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, London-New York, 1973.
[13] R.H. Lewis and P.F. Stiller, Solving the recognition problem for six lines using the Dixon resultant, Mathematics and Computers in Simulation 49 8(1999), 49(3):205-220.
[14] R.H. Lewis and S. Bridgett, Conic tangency equations and Apollonius problems in biochemistry and pharmacology, Mathematics and Computers in Simulation 8(2003), 61(2):101-114.
[15] R.H. Lewis and E.A. Coutsias, Algorithmic search for flexibility using resultants of polynomial systems, Lecture Notes in Computer Science 4869, Springer 8(2007), 68-79.
[16] R.H. Lewis, Comparing acceleration techniques for the Dixon and Macaulay resultants (abstract only), ACM Communications in Computer Algebra 8(2008), 42(1-2): 79-81.
[17] F.S. Macaulay, Some formulae in elimination, Proc. London Math. Soc. 35(1902), 3-27.
[18] F.S. Macaulay, Note on the resultant of a number of polynomials of the same degree, Proc. London Math. Soc. 21 (1921), 14-21.
[19] O. Ore, Formale theorie der linearen differentialgleichungen, I, J. Reine Angew. Math. 167 (1932), 221-234.
[20] C. H. Riquier, Les Systemes d' Equations aux Derivees Partielles, Gauthier-Villars, Paris, 1910.
[21] J. F. Ritt, Differential Equations from the Algebraic Standpoint, Coll. Publ. 14, Amer. Math. Soc., New York, 1932.
[22] J. F. Ritt, Differential Algebra, Coll. Publ. 33, Amer. Math. Soc., New York, 1950.