Linear Chevalley estimates

Abstract:

A Chevalley estimate for a germ of an analytic mapping $f$ is a function $l:\mathbb {N} \to \mathbb {N}$ such that if the composite with $f$ of a germ of an analytic function on the target vanishes to order at least $l(k)$, then it vanishes on the image to order at least $k$. Work of Izumi revealed the equivalence between regularity of a mapping (in the sense of Gabrielov, see $\S 1$) and the existence of a linear Chevalley estimate $l(k)$. Bierstone and Milman showed that uniformity of the Chevalley estimate is fundamental to several analytic and geometric problems on the images of mappings. The central topic of this article is uniformity of linear Chevalley estimates for regular mappings. We first establish the equivalence between uniformity of a linear Chevalley estimate and uniformity of a "linear product estimate" on the image: A linear product estimate on a local analytic ring (or, equivalently, on a germ of an analytic space) means a bound on the order of vanishing of a product of elements which is linear with respect to the sum of the orders of its factors. We study the linear product estimate in the central case of a hypersurface (i.e., the zero set of an analytic function). Our results show that a linear product estimate is equivalent to an explicit estimate concerning resultants. In the special case of hypersurfaces of multiplicity $2$, this allows us to prove uniformity of linear product estimates.