Polynomials in Several Variables

Polynomials are basic objects in mathematics. They have many applications in various fields of mathematics. One of the most important statements in the theory of polynomials is the Hilbert Basis Theorem, proved by David Hilbert in 1888, which states that every polynomial ring in several variables over a Noetherian ring is also Noetherian. In particular, when polynomials are considered over a field, the Hilbert Basis Theorem states that each ideal of the polynomial ring in several variables is finitely generated. The theory of Grobner bases for polynomials in several variables was developed by Bruno Buchberger in 1965, who named it on the honor of his adviser, Wolfgang Grobner. The importance of Grobner bases shows the facts that many fundamental problems in mathematics can be reduced by simple algorithms for constructing Grobner bases. This chapter considers the special subring of the ring of all polynomials in n variables which form symmetric polynomials.