Grobner Bases

The origins of the mathematics in this book date back more than two thou- sand years, as can be seen from the fact that one of the most important algorithms presented here bears the name of the Greek mathematician Eu- clid. The word &quote;algorithm&quote; as well as the key word &quote;algebra&quote; 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 Uzbek- istan and worked in Baghdad at the court of Harun al-Rashid's son. The word &quote;algorithm&quote; is actually a westernization of al-Khowarizmi's name, while &quote;algebra&quote; derives from &quote;al-jabr,&quote; a term that appears in the title of his book Kitab al-jabr wa'l muqabala, where he discusses symbolic methods for the solution of equations. This close connection between algebra and al- gorithms 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.