On the Index and the Order of Quasi-regular Implicit Systems of Differential Equations

2008/04/30 by Lisi D’Alfonso, Lisi D'Alfonso, Gabriela Jeronimo +7
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #Commutative Algebra (math.AC) #FOS: Mathematics #Numerical methods for differential equations #Polynomial and algebraic computation #math.AC #math.CA

paper · pdf · doi:10.48550/arxiv.0804.4842

openalex publication_date 2008/04/30 · arxiv created 2008/11/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is mainly devoted to the study of the differentiation index and the order for quasi-regular implicit ordinary differential algebraic equation (DAE) systems. We give an algebraic definition of the differentiation index and prove a Jacobi-type upper bound for the sum of the order and the differentiation index. Our techniques also enable us to obtain an alternative proof of a combinatorial bound proposed by Jacobi for the order. As a consequence of our approach we deduce an upper bound for the Hilbert-Kolchin regularity and an effective ideal membership test for quasi-regular implicit systems. Finally, we prove a theorem of existence and uniqueness of solutions for implicit differential systems.

Citations

• Die Frage der endlich vielen Schritte in der Theorie der Polynomideale
• The index of general nonlinear DAEs
• Jacobi's bound and normal forms computations. A historical survey

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