---
title: "On the fibers and semi-algebraicity of ReLU neuromanifolds"
source: "arXiv:2606.02826v1"
authors: "Axel Flinth, Stefano Mereta, Michele Pernice"
affiliation: "KTH Royal Institute of Technology / WASP"
year: 2026
category: "math.AG"
published: "2026-06-01"
---

# On the fibers and semi-algebraicity of ReLU neuromanifolds

**Authors**: Axel Flinth, Stefano Mereta, Michele Pernice
**arXiv**: 2606.02826v1 [math.AG]
**Published**: 2026-06-01

## Abstract

We study the semi-algebraicity of the neuromanifold M_d of a feedforward ReLU neural network and its symmetries. We prove that M_d is not a semi-algebraic quotient of the space of weights. We introduce honest open subsets where the network shows no hidden symmetries, conjecture the maximal honest open is always semi-algebraic, and prove it is Zariski open in the shallow case.

## Key Concepts
- [[neuromanifold|Neuromanifold]] — function space parametrized by network weights
- [[neuroalgebraic-geometry|Neuroalgebraic Geometry]] — algebro-geometric study of neural networks
- [[semi-algebraic-set|Semi-algebraic Set]] — sets defined by polynomial equalities/inequalities
- [[honest-open-subset|Honest Open Subset]] — region free of hidden symmetries
- [[hidden-symmetries-neural|Hidden Symmetries]] — symmetries beyond scaling and permutation
- [[parametrization-map|Parametrization Map]] — weight-to-function mapping
- [[scaling-permutation-symmetry|Scaling & Permutation Symmetries]] — trivial NN symmetries
- [[fiber-of-parametrization|Fiber of Parametrization]] — preimage of a function
