GROBNER BASES AND INVOLUTIVE METHODS FOR ALGEBRAIC AND DIFFERENTIAL-EQUATIONS

In this paper, we consider and illustrate by examples some recently de veloped computer algebra methods for analysing and solving nonlinear a lgebraic and differential equations. The foundation of these methods i s either the transformation of the initial equations to an equivalent, often called standard, form or their reduction to a finite set of sub systems in standard form. As a standard form we consider various Grobn er bases with special emphasis on its involutive extension. Applicatio ns to the symmetry and integrability analysis of partial differential equations as well as to solving systems of polynomial equations are di scussed.