Unipotent differential algebraic groups as parameterized differential Galois groups
parameterized differential Galois theory
The parametrized Picard-Vessiot (PPV) Galois theory is a Galois theory of linear differential equations, with respect to a derivation \(\partial\), of the form \(\partial Y=A Y\), where A is a squared matrix whose entries belong to a \(\{\partial\}\cup \Delta\)-differential field \(k\), with \(\{\partial\}\cup\Delta\) a finite set of commuting derivations. This theory was presented in [\textit{P. J. Cassidy} and \textit{M. F. Singer}, IRMA Lect. Math. Theor. Phys. 9, 113--155 (2007; Zbl 1356.12004)], where the next example is given: \(\frac{dy}{dx}=\frac{x}{t} y\).NEWLINENEWLINEA PPV extension \(K\) of \(k\) associated with \(\partial Y=A Y\) is a \(\{\partial\}\cup \Delta\)-differential field verifying:NEWLINENEWLINE1. \(K=k\langle Z\rangle\) is generated as a \(\{\partial\}\cup \Delta\)-differential field by the entries of a matrix \(Z\) such that \(\partial Z=AZ\).NEWLINENEWLINE2. The field of constants of \(k\) and \(K\) with respect to \(\partial\) coincide, let as call it \(C\).NEWLINENEWLINEIt was proved in the previous paper that \(C\) is a \(\Delta\)-field and if it is differentially closed the PPV extension always exists and it is unique up to \(\{\partial\}\cup \Delta-k\)-isomorphisms.NEWLINENEWLINENEWLINEThe PPV Galois theory studies the group of differential symmetries of the PPV extension, called a parameterized differential Galois group (PPV group). These are linear differential algebraic groups (LDAGs), groups of matrices whose entries satisfy polynomial differential equations in the parameters.NEWLINENEWLINENEWLINEThe paper under review deals with some aspect of the inverse and direct problems of PPV Galois theory over \(k=\mathcal{U} (x)\), where \(\mathcal{U}\) is a universal differential field with derivations \(\Delta\). It gives a characterization of those LDAGs that occur as PPV groups over \(k=\mathcal{U} (x)\). Namely, a LDAG is a PPV group if and only if it constains a finitely generated Kolchin-dense subgroup. Related with the direct problem two algorithms are presented:NEWLINENEWLINENEWLINEAlgorithm 1: Computes the defining equations of the PPV group \(G\) of \(\partial Y=A Y\) assuming that \(G\) has differential type \(0\) (in a certain sense finite-dimensional).NEWLINENEWLINENEWLINEAlgorithm 2: Determines if the PPV group \(G\) of \(\partial Y=A Y\) has the property that \(G/R_u(G)\) is 'constant', where \(R_u(G)\) is the unipotent radical of \(G\).NEWLINENEWLINECombining both algorithms allows determining if \(G/R_u(G)\) is constant (as defined in the paper) and if so to compute the defining equations of \(G\). If \(G/R_u(G)\) is not necessarily constant, an algorithm that computes \(G/R_u(G)\) is given in [the authors, Int. Math. Res. Not. 2015, No. 7, 1733--1793 (2015; Zbl 1339.12003)].
scientific article; zbMATH DE number 6349676
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