Holomorphy rings and the Nullstellensatz over large fields
Abstract
For an infinite field K and an ideal I⊆ K[X1,…,Xn], let J(ZK(I)) be the ideal of polynomials vanishing on the K-rational zero locus of I. A Nullstellensatz over K is an algebraic description of this closure operator on ideals. Hilbert's classical theorem provides such a description via the radical of I over any algebraically closed field. In the present article, we introduce a unifying framework for Nullstellensätze over arbitrary rings using the notion of an A-radical, for A is a subring of K(X) containing K[X]. We show that a single canonical holomorphy ring gives a Nullstellensatz over every infinite field, recovering Hilbert's theorem when K is algebraically closed. In addition, we give a new characterization of large fields (in the sense of Pop) via a particular Nullstellensatz: a perfect field is large exactly when the \emph{rational} holomorphy ring An(K)--the intersection of all K-rational places of residue field exactly K--satisfies the corresponding Nullstellensatz. Finally we study the order structure of the class of rings A for which the A-radical gives a Nullstellensatz. We show that over non-algebraically closed fields this class has no minimal element, showing there is no optimal Nullstellensatz over such fields.
Related
- Strong and Explicit Forms of the One-Sided Nullstellensatz over Division Rings
- Short arithmetic terms for the cardinality of elliptic curves over rings of remainder classes and over finite fields
- The Influence of Large-Scale Weather Patterns on Potential Wind Energy Fields over Mexico
- Peskine and Szpiro's Dimension inequality holds over local rings of depth at most one
- Automorphisms of Twisted Chevalley Groups of type 2 A3 and 2 A4 over Local Rings
BibTeX
@misc{kovacsics2026holomorphy,
title = {{Holomorphy rings and the Nullstellensatz over large fields}},
author = {Pablo Cubides Kovacsics and Sebastián Uribe Rincón},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21039},
eprint = {2610.21039},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21039},
}
RIS
TY - UNPB
TI - Holomorphy rings and the Nullstellensatz over large fields
AU - Pablo Cubides Kovacsics
AU - Sebastián Uribe Rincón
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21039
UR - https://arxiv.org/abs/2610.21039
AB - For an infinite field K and an ideal I⊆ K[X_1,…,X_n], let J(Z_K(I)) be the ideal of polynomials vanishing on the K-rational zero locus of I. A Nullstellensatz over K is an algebraic description of this closure operator on ideals. Hilbert's classical theorem provides such a description via the radical of I over any algebraically closed field. In the present article, we introduce a unifying framework for Nullstellensätze over arbitrary rings using the notion of an A-radical, for A is a subring of K(X) containing K[X]. We show that a single canonical holomorphy ring gives a Nullstellensatz over every infinite field, recovering Hilbert's theorem when K is algebraically closed. In addition, we give a new characterization of large fields (in the sense of Pop) via a particular Nullstellensatz: a perfect field is large exactly when the \emph{rational} holomorphy ring A_n(K)--the intersection of all K-rational places of residue field exactly K--satisfies the corresponding Nullstellensatz. Finally we study the order structure of the class of rings A for which the A-radical gives a Nullstellensatz. We show that over non-algebraically closed fields this class has no minimal element, showing there is no optimal Nullstellensatz over such fields.
ER -