Gröbner bases and involutive methods for algebraic and differential equations
type: Item
instance of: scholarly article
title: Gröbner bases and involutive methods for algebraic and differential equations
zbMATH Open document ID: 0904.13014
DOI: 10.1016/S0895-7177(97)00060-5
published in: Mathematical and Computer Modelling
publication date: 1999-01-04T00:00:00Z
review text: The paper illustrates computer algebra methods for analyzing systems of (non)linear algebraic partial differential equations. Sections 2 and 3 present background to and the algebraic Gröbner bases (GB-)algorithm and corresponding differential analogs; difficulties occurring in the latter are frequently non-terminating algorithms in the case of generation of differential ideals by nonlinear polynomials \(f_l=f_l(x,u,\ldots, D^\alpha u)\), \(l=1,\ldots, k\). Section 4 discusses involutive algebraic bases. For definiteness assume the monomials of \(K[x_1,\ldots, x_n]\) compared by degree reverse lex ordering, as customary in GB-theory. Then \(x_1\succ x_2 \succ \ldots \succ x_n\) and we have the notion of a leading term lt\((p)\) of a polynomial \(p\). For a term \(u=c\cdot x_1^{k_1} \cdots x_i^{k_i}\), in which all the first \(i\) variables genuinely occur, the set of \textit{multiplicative} variables is \({\mathcal M}(u)= \{x_i, x_{i+1},\ldots, x_n\}.\) Given polynomials \(p\) and \(q\), assume \(p\) has a term \(t\) such that \(t=\text{lt}(q)\times u\) with \(u\) variables in \({\mathcal M}({\text lt}(q))\). Then \(p\) can be involutively reduced by \(q\) yielding a reduction \(p\rightarrow p'=p-q\times u\). Similarly as in GB-theory one can define this way the involutive normal form \(N_I(p,F)\) of a polynomial \(p\) modulo a finite polynomial set \(F\) and develop in favorable cases (e.g. zero dimensional ideals) a theory of involutive bases [see \textit{A. Yu. Zharkov} and \textit{Yu. A. Blinkov} Math. Comput. Simulation 42, 323-332 (1996) and \textit{A. Yu. Zharkov} in: Algorithms in Algebraic Geometry and Applications, Proc. MEGA-94 Conf., Santander 1994, Prog. Math. 143, 389-399 (1996; Zbl 0878.13019)]. Section 5: Involutive bases are (if existent) special cases of GBs with particular properties. These make them attractive for studies of linear PDE-systems by means of the correspondence \(\frac{\partial^{\alpha_1+\ldots +\alpha_n} u_j} {\partial x_1^{\alpha_1}…
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13N10
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 35C05
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1067235
zbMATH Keywords: Gröbner bases
zbMATH Keywords: involutive bases
zbMATH Keywords: differential algebra
zbMATH Keywords: partial differential equations
zbMATH Keywords: Lie methods in differential equations
author: Vladimir Gerdt
reviewed by: Alexander Kovačec
MaRDI profile type: Publication
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full work available at URL: s0895-7177(97)00060-5
OpenAlex ID: W1971064965
description: scientific article; zbMATH DE number 1067235