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:   
Talk
Number:
04-07  E-mail:    
Session: 4- Elimination Theory and Applications Schedule:
 
Room:
Saturday, 14:00
 
B-2620
Related website:  
Title of
presentation:
An Attempt to Apply Dixon Resultant to Differential Elimination
Abstract:

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].

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