Editing Elimination theory (section)

==History and connection to modern theories==
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 equation]]s and [[abelian group]] with [[Hermite normal form]] and [[Smith normal form]].

Before the 20th century, different types of ''eliminants'' were introduced, including ''[[resultant]]s'', and various kinds of ''[[discriminant]]s''. In general, these eliminants are also [[invariant theory|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". Nevertheless [[Hilbert's Nullstellensatz]], may be considered to belong to elimination theory, as it asserts that a system of polynomial equations does not have any solution if and only if one may eliminate all unknowns to obtain the constant equation 1 = 0.

Elimination theory culminated with the work of [[Leopold Kronecker]], and finally  [[Francis Sowerby Macaulay|Macaulay]], who introduced [[multivariate resultant]]s and [[Resultant#U-resultant|U-resultants]], providing complete elimination methods for systems of polynomial equations, which are described in the chapter on ''Elimination theory'' in the first editions (1930) of [[Bartel Leendert van der Waerden|van der Waerden's]] ''[[Moderne Algebra]]''.

Later, elimination theory was considered old-fashioned and removed from subsequent editions of ''Moderne Algebra''. It was generally ignored until the introduction of [[computer]]s, and more specifically of [[computer algebra]], which again made relevant the design of efficient elimination algorithms, rather than merely existence and structural results. The main methods for this renewal of elimination theory are [[Gröbner bases]] and [[cylindrical algebraic decomposition]], introduced around 1970.