Algebraic normalisation
Adam Białożyt
TL;DR
The paper develops a purely geometric, $c$-algebraic analogue of analytic normalisation for algebraic sets by introducing $c$-algebraic functions, $a$-normal sets, and the $a$-normalisation. It constructs a universal denominator to obtain representations of $f\in\mathcal{O}_c^a(A)$ as rational functions and builds a canonical $a$-normalisation as the graph of a generating set, enabling a direct proof of a Nullstellensatz in this setting. It analyzes the relationship between the canonical $a$-normalisation and the growth exponent of $c$-algebraic functions, and discusses connections to seminormalisation in algebraic geometry. The framework yields a transparent, global approach to normalization in the $c$-algebraic context and provides tools for finite generation of $\mathcal{O}_c^a(A)$ as a module over $\mathbb{C}[z]$, with a streamlined Nullstellensatz proof accompanying it.
Abstract
We present a strictly geometric c-algebraic version of the analytic set normalisation. With the introduced tool we prove the Nullstellensatz for c-algebraic functions and study the growth exponent of a c-algebraic function.