Nonlinear Science Theory and Applications Series editor Arun V. Holden, Centre for Nonlinear Studies, University of Leeds. Editorial Board Shun Ichi Amari, Tokyo Peter L. Christiansen, Houston David Crighton, Cambridge Robert Helleman, Houston David Rand, Warwick J. C. Roux, Bordeaux Introduction to the algebraic theory of invariants of differential equations K. S. Sibirsky This monograph considers polynomial invariants and comitants of autonomous systems of differential equations with right-hand sides relative to various transformation groups of the phase space. Some questions connected with the construction of polynomial bases and complete systems of invariants and comitants are investigated and many applications to the qualitative theory of differential equations are indicated. The two-dimensional system with quadratic right-hand sides is investigated in detail. For such systems, polynomial bases of affine comitants as well as of polynomial syzygies are constructed. Polynomial bases and complete systems of invariants relative to orthogonal transformations and rotations of the phase plane are also investigated. The results are applied in determination of the symmetry axes, necessary and sufficient conditions for the existence of a centre and of an isochronous centre, computation of the cyclicity of a focus. This book will be of interest to undergraduate as well as postgraduate students and to researchers in mathematics and mechanics.