Testing Algebraic Complete Intersections
Abstract
Given independent and identically distributed samples samples from a probability distribution in a potentially high-dimensional real space, we study the problem of testing whether the distribution is concentrated near a real algebraic complete intersection of prescribed dimension, bounded degree, and bounded condition number. We design an explicit and effective learning procedure which either certifies the nonexistence of such a manifold, up to a controlled relaxation of the approximation threshold, or returns a candidate regression manifold with controlled geometric complexity. Equivalently, the procedure tests the manifold hypothesis within this hypothesis class. The proposed procedure relies on quantitative geometric estimates for regular polynomial systems, which lead to a tractable auxiliary optimization problem. We then develop a data-driven algorithm to solve this auxiliary optimization problem, establishing explicit bounds on its sample and arithmetic complexity.
Related
- Triple intersections for algebraic curves on the real torus
- K-stability and birational rigidity of Fano 3-fold weighted complete intersections
- Existence of weak Landau-Ginzburg models for weighted complete intersections
- Generic Triviality of Linear Automorphisms of Complete Intersections
- Some Remarks on Regular Sequences and Complete Intersections
BibTeX
@misc{tamai2026testing,
title = {{Testing Algebraic Complete Intersections}},
author = {Alessandro Tamai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12288},
eprint = {2610.12288},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12288},
}
RIS
TY - UNPB
TI - Testing Algebraic Complete Intersections
AU - Alessandro Tamai
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12288
UR - https://arxiv.org/abs/2610.12288
AB - Given independent and identically distributed samples samples from a probability distribution in a potentially high-dimensional real space, we study the problem of testing whether the distribution is concentrated near a real algebraic complete intersection of prescribed dimension, bounded degree, and bounded condition number. We design an explicit and effective learning procedure which either certifies the nonexistence of such a manifold, up to a controlled relaxation of the approximation threshold, or returns a candidate regression manifold with controlled geometric complexity. Equivalently, the procedure tests the manifold hypothesis within this hypothesis class. The proposed procedure relies on quantitative geometric estimates for regular polynomial systems, which lead to a tractable auxiliary optimization problem. We then develop a data-driven algorithm to solve this auxiliary optimization problem, establishing explicit bounds on its sample and arithmetic complexity.
ER -