Problems which arise naturally in computer algebra can be considered as geometric dominance queries in which we are required to identify only one point (if any) which lies within the query range. In this paper, we consider the differences between this problem and other more traditional range query problems. In particular we investigate the inherent complexity of this problem and data structures which can be used for answering this query.

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 Gröbner 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 Gröbner basis for the ideal generated by F is bounded by $2(({{d^2} / 2}) + d)^{2^{n-1}}$.