» Becker T., Weispfenning V. - Groebner bases and commutative algebra
Becker T., Weispfenning V. - Groebner bases and commutative algebra
||Groebner bases and commutative algebra
||Becker T., Weispfenning V.
The origins of the mathematics in this book date back more than two thousand years, as can be seen from the fact that one of the most important algorithms presented here bears the name of the Greek mathematician Euclid. The word "algorithm" as well as the key word "algebra" in the title of this book come from the name and the work of the ninth-century scientist Mohammed ibn Musa al-Khowarizmi, who was born in what is now Uzbekistan and worked in Baghdad at the court of Harun al-Rashid's son. The word "algorithm" is actually a westernization of al-Khowarizmi's name, while "algebra" derives from "al-jabr," a term that appears in the title of his book Kitab al-jabr wa 'I muqabala, where he discusses symbolic methods for the solution of equations. This close connection between algebra and algorithms lasted roughly up to the beginning of this century; until then, the primary goal of algebra was the design of constructive methods for solving equations by means of symbolic transformations.
During the second half of the nineteenth century, a new line of thought began to enter algebra from the realm of geometry, where it had been successful since Euclid's time, namely, the axiomatic method. The starting point of the axiomatic approach to algebra is the question, What kind of object is a symbolic solution to an algebraic equation? To use a simple example, the question would be not only, What is a solution of ax + b = 0, but also, What are the properties of the objects a and b that allow us to form the object —b/al The axiomatic point of view is that these are objects in a surrounding algebraic structure which determines their behavior. The algebraic structure in turn is described and determined by properties that are laid down in a set of axioms.
The foundations of this approach were laid by Richard Dedekind, Ernst Steinitz, David Hilbert, Emmy Noether, and many others. The axiomatic method favors abstract, non-constructive arguments over concrete algorithmic constructions. The former tend to be considerably shorter and more elegant than the latter. Before the arrival of computers, this advantage more or less settled the question of which one of the two approaches was to be preferred: the algorithmic results of mathematicians like Leopold Kro-necker and Paul Gordan were way beyond the scope of what could be done with pencil and paper, and so they had little to offer except being more tedious them their non-constructive counterparts.
On the other hand, it would be a mistake to construe the axiomatic and