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Elimination theory
In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables.
==Connection to modern theories==
The linear case would now routinely be handled by Gaussian 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. In particular the discriminant appears in invariant theory, and is often constructed as the invariant of either a curve or an ''n''-ary ''k''-ic form. Whilst discriminants are always constructed resultants, the variety of constructions and their meaning tends to vary. A modern and systematic version of theory of the discriminant has been developed by Gelfand and coworkers. Some of the systematic methods have a homological basis, that can be made explicit, as in Hilbert's theorem on syzygies. This field is at least as old as Bézout's theorem.
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