Multiplicities of Noetherian deformations

TL;DR

This paper advances the theory of Noetherian functions by providing an effective upper bound on the local solutions of parameter-dependent systems, supporting conjectures about their local geometric complexity.

Contribution

It offers a new effective bound for solutions of Noetherian systems with parameters, even under degenerations, aiding the development of local geometric estimates.

Findings

Bound remains valid under system degenerations at psilon=0

Supports Khovanskii's conjecture on local geometry estimates

Leads to an effective Lojasiewicz inequality for Noetherian functions

Abstract

The \emph{Noetherian class} is a wide class of functions defined in terms of polynomial partial differential equations. It includes functions appearing naturally in various branches of mathematics (exponential, elliptic, modular, etc.). A conjecture by Khovanskii states that the \emph{local} geometry of sets defined using Noetherian equations admits effective estimates analogous to the effective \emph{global} bounds of algebraic geometry. We make a major step in the development of the theory of Noetherian functions by providing an effective upper bound for the local number of isolated solutions of a Noetherian system of equations depending on a parameter $\epsilon$, which remains valid even when the system degenerates at $\epsilon=0$. An estimate of this sort has played the key role in the development of the theory of Pfaffian functions, and is expected to lead to similar results in the Noetherian setting. We illustrate this by deducing from our main result an effective form of the Lojasiewicz inequality for Noetherian functions.