Elimination theory

In algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables.

The linear case would now routinely be handled by Gauss-Jordan elimination, rather than the theoretical solution provided by Cramer s rule. In the same way, computational techniques for elimination can in practice be based on Gröbner basis methods. There is however older literature on types of eliminant , including resultants to find common roots of polynomials, discriminants and so on. Some of the systematic methods have a homological algebra basis, that can be made explicit, as in Hilbert s syzygy theorem. This field is at least as old as Bézout s theorem.

The historical development of and effectively linearised while dropping the explicit constructive content. The process continued over many decades: the work of F.S. Macaulay who gave his name to Cohen-Macaulay modules was motivated by elimination.

There is also a logical content to elimination theory, as seen in the .