Effective bounds for the consistency of differential equations. (November 2018)

• Record Type: Journal Article
• Authors: Gustavson, Richard; León Sánchez, Omar
• Is Part Of: Journal of symbolic computation. Volume 89(2018)
• Page Start: 41
• Page End: 72
• Publication Date: 2018-11
• Subjects: 12H05 -- 14Q20 -- 35G50; Algebraic differential equations -- Antichain sequences -- Hilbert–Samuel function; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2017.11.003
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: One method to determine whether or not a system of partial differential equations is consistent is to attempt to construct a solution using merely the "algebraic data" associated to the system. In technical terms, this translates to the problem of determining the existence of regular realizations of differential kernels via their possible prolongations. In this paper we effectively compute an improved upper bound for the number of prolongations needed to guarantee the existence of such realizations, which ultimately produces solutions to many types of systems of partial differential equations. This bound has several applications, including an improved upper bound for the order of characteristic sets of prime differential ideals. We obtain our upper bound by proving a new result on the growth of the Hilbert–Samuel function, which may be of independent interest.