A theorem on refining division orders by the reverse lexicographic order
If \(S_ d\) is the K-vector space of homogeneous polynomials of degree d in \(S=k[x_ 1,...,x_ n]\), where K is an infinite field, then a (partial) order \(>\) on the monomials of \(S_ d\) for each d is compatible with the monoid structure of the monomials of S if \(x^ A>x^ B\) implies \(x^ Cx^ A>x^ Cx^ B\) for all monomials \(x^ C\). If the order \(>\) is strict, then it can be used in applying the division algorithm to construct a standard basis for homogeneous ideals I whose properties vary with I. Many computaions of this type lead to results which are quite interesting from the combinatorial viewpoint as is illustrated by the algebraic theory of shellable posets, for example. In this paper the interest is more directly algebro-geometric. In particular the main theorem (theorem 6) is as follows: If \(>_ 3\) is the refinement of an (arbitrary) order \(>_ 1\) by the reverse lexicographic order \(>\) (rlex), and if \(I\subset S\) is a homogeneous ideal, then there exists a Zariski open subset \(U\subset GL(n)\) such that for any \(g\in U\), \(in_ 1(g\cdot U)\) and \(in_ 3(g\cdot I)\) have the same regularity. As noted, the reverse lexicographic order is often a most efficient choice for constructing standard bases when one can make a free choice of order, as in the computation of syzygies or Hilbert functions. These latter computations have important consequences in combinatorics also. Theorem 6 carries some of these efficiency properties of reverse lexicographic order to more general settings, which will be useful in a variety of applications.
scientific article; zbMATH DE number 4039965