A c-holomorphic effective Nullstellensatz with parameter

TL;DR

This paper establishes a parameter-uniform effective Nullstellensatz for families of c-holomorphic functions, providing bounds on zero cycles that are independent of the parameter, under certain intersection conditions.

Contribution

It proves a local Nullstellensatz with a parameter for c-holomorphic functions with an effective exponent, removing the proper intersection assumption for isolated zeroes.

Findings

Effective exponent independent of the parameter

Nullstellensatz holds for families with isolated zeroes

Zero cycle degree bounds are established

Abstract

We prove a local Nullstellensatz with parameter for a continuous family of c-holomorphic functions with an effective exponent independent of the parameter: the local degree of the cycle of zeroes of the central section section. We assume that this central section defines a proper intersection and we show that we can omit this assumption in case of isolated zeroes.