The structure of polynomial ideals and Gröbner bases

This paper introduces the cone decomposition of a polynomial ideal. It is shown that every ideal has a cone decomposition of a standard f orm. Using only this and combinatorial methods, the following sharpened bound for the degree of polynomials in a Grobner basis can be produced. Let $K[x_{1},\cdots , x_{n}]$ be a ring of multivariate polynomials with coefficients in a field $K$, and let $F$ be a subset of this ring such that $d$ is the maximum total degree of any polynomial in $F$. Then for any admissible ordering, the total degree of polynomials in a Grobner basis for the ideal generated by $F$ is bounded by $2(({{d^2} / 2}) + d)^{2^{n-1}}$.