Grِbner bases and involutive methods for algebraic and differential equations

In this paper, we consider and illustrate by examples some recently developed computer algebra methods for analyzing and solving nonlinear algebraic and differential equations. The foundation of these methods is either the transformation of the initial equations to an equivalent, often called standard, form or their reduction to a finite set of subsystems in standard form. As a standard form we consider various Grِbner bases with special emphasis on its involutive extension. Applications to the symmetry and integrability analysis of partial differential equations as well as to solving systems of polynomial equations are discussed.