This is an introductory paper on elimination methods applicable
to a system of nonlinear polynomials equations. We discuss three
different approaches for solving a system of nonlinear polynomial
equations. The first approach of resultants is based on the
theory of determinants, and was developed in the late 19th century and
the early 20th century. The main idea is to generate from a
system of nonlinear polynomial equations, a possibly larger
system of independent polynomial equations such that there are as
many equations as terms in the polynomials so that each term can
be used as an unknown and the theory of linear system of
equations can be applied. We describe the resultants of
Sylvester, Bezout, Dixon's extension of Bezout's resultant, and
Macaulay's resultant, followed by some recent extensions of
The second approach is based on polynomial ideal theory and generates special bases of polynomial ideals, called Groebner bases. An algorithm for computing such bases was given by Buchberger and is described here. We review some recent developments in the Groebner basis theory using which nonlinear polynomial equations with finitely many solutions can be efficiently solved.
The third approach is based on Ritt's characteristic set construction, motivated by the analysis and decomposition of the zero sets of systems of polynomial equations. This approach has been recently popularized by Wu Wen-tsun who has impressively demonstrated its application to automated geometry theorem proving. Using Wu's method, it is possible to automatically prove, in a matter of seconds, nontrivial theorems in plane Euclidean geometry which human experts find difficult to prove.
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