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Concept
Elimination theory
Summary
In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables, in order to solve systems of polynomial equations.
Classical elimination theory culminated with the work of Francis Macaulay on multivariate resultants, as described in the chapter on Elimination theory in the first editions (1930) of Bartel van der Waerden's Moderne Algebra. After that, elimination theory was ignored by most algebraic geometers for almost thirty years, until the introduction of new methods for solving polynomial equations, such as Gröbner bases, which were needed for computer algebra.
The field of elimination theory was motivated by the need of methods for solving systems of polynomial equations.
One of the first results was Bézout's theorem, which bounds the number of solutions (in the case of two polynomials in two variables at Bézout time).
Except for Bézout's theorem, the general approach was to eliminate variables for reducing the problem to a single equation in one variable.
The case of linear equations was completely solved by Gaussian elimination, where the older method of Cramer's rule does not proceed by elimination, and works only when the number of equations equals the number of variables. In the 19th century, this was extended to linear Diophantine equations and abelian group with Hermite normal form and Smith normal form.
Before the 20th century, different types of eliminants were introduced, including resultants, and various kinds of discriminants. In general, these eliminants are also invariant under various changes of variables, and are also fundamental in invariant theory.
All these concepts are effective, in the sense that their definitions include a method of computation. Around 1890, David Hilbert introduced non-effective methods, and this was seen as a revolution, which led most algebraic geometers of the first half of the 20th century to try to "eliminate elimination".
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