30 - Reverse Lexicographic Initial Ideals of Generic Ideals are Finitely Generated

Summary

Abstract

This article generalizes the well-known notion of generic forms to the algebra R′, introduced in [12]. For the total degree, then reverse lexicographic order, we prove that the initial ideal of an ideal generated by finitely many generic forms (in countably infinitely many variables) is finitely generated. This contrasts to the lexicographic order, for which initial ideals of generic ideals in general are non-finitely generated.

The natural question, “is the reverse lexicographic initial ideal of an homogeneous, finitely generated ideal in R′ finitely generated” is posed, but not answered; we do, however, point out one direction of investigation that might provide the answer: namely to view such an ideal as the “specialization” of a generic ideal.

Introduction

In this article, we study the initial ideals of generic and “almost generic” ideals with respect to the (total degree, then) reverse lexicographic term order. For a generic ideal I ⊂ K[x₁,…, x_n], generated by r ≤ n forms, there is a well-known conjecture on how gr(I) looks like. In particular, gr(I) is minimally generated in K[x₁,…, x_r]. We interpret this result in the setting of the ring R′, introduced in [12]: this ring, which is a proper subring on the power series ring on countably many variables, and which properly contains the polynomial ring on the same set of indeterminates, is the habitat of “generic forms in countably many indeterminates”.