Solving Polynomial Equation Systems

In this fourth and final volume the author extends Buchberger''s Algorithm in three different directions. First he extends the theory to group rings and other Ore-like extensions and provides an operative scheme that allows one to set a Buchberger theory over any effective associative ring. Second he covers similar extensions as tools for discussing parametric polynomial systems the notion of SAGBI-bases Gröbner bases over invariant rings and Hironaka''s theory. Finally Mora shows how Hilbert''s followers - notably Janet Gunther and Macaulay - anticipated Buchberger''s ideas and discusses the most promising recent alternatives by Gerdt (involutive bases) and Faugère (F4 and F5). This comprehensive treatment in four volumes is a significant contribution to algorithmic commutative algebra that will be essential reading for algebraists and algebraic geometers.