Application of algebras to differential systems. Ed. by N. I. Vulpe. (Q2744152)

The monograph deals with the groups of linear transformations of phase variables for the first-order polynomial systems of autonomous differential equations. The main goal is to examine their linear representations in the space of coefficients. For this purpose, two types of algebras are constructed and studied: finite dimensional Lie algebras of operators for groups of representations and finitely determined graded algebras of comitants and invariants of differential systems for the group of unimodular transformations. For the latter algebras, the minimal number of generators, their types, defining relations for generators (syzygies), and their Krull dimension are discussed by exploiting Hilbert series. Using Lie algebras of operators and appropriate algebras of comitants and invariants, the author computes dimensions of orbits and finds invariant integrals for specific systems of differential equations for different groups of linear transformations of phase variables.NEWLINENEWLINENEWLINEThe book contains known results along with those obtained recently by the author and his colleagues and its materials have been used for the course the author has presented to mathematics students at the Tiraspol State University. It may be of interest to researchers and graduate students in mathematics and physics.