# Weak and semi normalization in real algebraic geometry

## TL;DR

This paper introduces weak-normalization and seminormalization concepts in real algebraic geometry, relating them to rings of rational functions and providing characterizations and descriptions of specific varieties.

## Contribution

It defines and explores weak-normalization and seminormalization relative to the central locus, offering new algebraic and geometric characterizations in real algebraic geometry.

## Key findings

- Characterization of varieties via rings of rational functions
- Full description of centrally seminormal curves
- Connections between normalization concepts and geometric properties

## Abstract

We define the weak-normalization and the seminormalization of a real algebraic variety relative to its central locus. The study is related to the properties of the rings of continuous rational functions and hereditarily rational functions on real algebraic varieties. We provide in particular several characterizations (algebraic or geometric) of these varieties, and provide a full description of centrally seminormal curves.