Solving Polynomial Equation Systems IV: Volume 4, Buchberger Theory and Beyond: 158 (Encyclopedia of Mathematics and its Applications, Series Number 158) - Hardcover

'The author treats all relevant steps and results in great detail also including advanced and most recent developments respectively, both of the theoretical and the algorithmic side ... an abundance of worked out examples shows the effectivity of the various algorithms.' Monatshefte fur Mathematik "The material contained in the book is remarkably wide-ranging and includes the most recent developments in the field. The book is ... a fundamental reference for anyone from undergraduate students to researchers interested in (computational aspects of) algebra." Mathematical Reviews "I have to admit that I fell in love with this book at first sight; for it is not just extremely well organized, it is also written in a style that is a joy to read... To sum up, this is a wonderful book, beautifully written and produced, that should be in every mathematical library. Actually, if you are a serious user of Grobner bases you will probably wish to have your own copy of the book, which, I bet will soon be very well thumbed." S.C. Coutinho, SIGACT News"

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.