Infinite Versions of Hilbert's Nullstellensatz

A. Bernhard Zeidler

arxiv: 2502.10566 · v1 · submitted 2025-02-14 · 🧮 math.AC

Infinite Versions of Hilbert's Nullstellensatz

A. Bernhard Zeidler This is my paper

Pith reviewed 2026-05-23 02:51 UTC · model grok-4.3

classification 🧮 math.AC

keywords Hilbert Nullstellensatzinfinite variablespolynomial ringscommutative algebraideal theorystrong Nullstellensatzalgebraic geometry

0 comments

The pith

Hilbert's Nullstellensatz admits many equivalent formulations in infinite dimensions, and the strong version persists in large polynomial rings.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper assembles a collection of statements that are equivalent to Hilbert's Nullstellensatz once the underlying polynomial ring is allowed to contain infinitely many variables. It shows these equivalences hold without collapsing when the number of variables becomes infinite. It further establishes that the strong form of the theorem, relating the radical of an ideal to its vanishing set, continues to apply in sufficiently large such rings. This matters because algebraic geometry and ideal theory can then be used on systems whose variable count is not bounded in advance.

Core claim

There exists a long list of equivalent formulations of Hilbert's Nullstellensatz in infinite dimensions, and the strong Nullstellensatz persists in large polynomial rings.

What carries the argument

The list of equivalent formulations of the Nullstellensatz extended to polynomial rings in infinitely many variables, together with the persistence property of the strong form.

If this is right

The weak and strong Nullstellensatz remain equivalent statements even after the variable set is made infinite.
The strong Nullstellensatz applies verbatim to polynomial rings whose cardinality of variables exceeds any fixed bound.
Classical consequences of the Nullstellensatz, such as relations between ideals and varieties, transfer directly to the infinite-variable setting.
Proof techniques that rely on finite-variable reductions continue to function once the ring is large enough.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Infinite-variable versions could be used to model algebraic relations in function spaces or formal power series rings.
The persistence result suggests that other ideal-theoretic theorems might also survive passage to infinite dimensions under similar size conditions.
One could test the boundary by constructing rings whose variable set has intermediate cardinality between countable and continuum.

Load-bearing premise

The usual definitions of polynomial rings in infinitely many variables and the classical notions of weak and strong Nullstellensatz extend while preserving the listed equivalences and the persistence result without further restrictions.

What would settle it

An explicit ideal I in a polynomial ring over an infinite set of variables such that the radical of I fails to equal the ideal of all polynomials vanishing on the common zeros of I.

read the original abstract

We compile a long list of equivalent formulations of Hilbert's Nullstellensatz in infinite dimensions, and prove a persistence result for the strong Nullstellensatz in large polynomial rings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper compiles a list of equivalent formulations of the Nullstellensatz for infinite-variable polynomial rings and proves persistence of the strong version, which organizes existing ideas but adds limited new depth.

read the letter

The core of this paper is a compilation of many equivalent statements of Hilbert's Nullstellensatz in the setting of polynomial rings over infinitely many variables, plus a proof that the strong form persists in sufficiently large such rings. That persistence result is the part presented as new. Bringing the equivalents together in one place can be convenient for people who work with infinite indeterminates, since different characterizations sometimes fit better with different arguments or constructions. The approach appears to rely on standard reductions to finitely supported polynomials, which is the usual way these extensions are handled in commutative algebra. The stress-test note aligns with that: no obvious circularity or definitional mismatch shows up from the abstract. The definitions of the rings and the weak/strong versions seem to carry over directly without extra restrictions, at least on the surface. Soft spots are minor but real. The abstract gives no proof sketches, so it is impossible to check whether the equivalences are all straightforward or if any requires non-trivial work in the infinite case. If the persistence proof is essentially an inductive or limit argument from the finite-variable theorem, then the novelty is mostly organizational rather than conceptual. The term large polynomial rings also needs a precise definition to avoid vagueness about what extensions are allowed. This is a paper for commutative algebraists or algebraic geometers who already deal with infinite-variable rings and want a reference list of Nullstellensatz forms. A reader looking for a consolidated statement of known equivalents plus one stability result would find it useful. It deserves a serious referee because the claims are the sort that can be checked directly against the literature and the definitions, even if the overall advance is modest. I would send it to peer review rather than desk reject.

Referee Report

0 major / 3 minor

Summary. The manuscript compiles a long list of equivalent formulations of Hilbert's Nullstellensatz for polynomial rings in infinitely many variables and proves a persistence result for the strong Nullstellensatz in large polynomial rings.

Significance. If the equivalences and persistence hold under the standard extensions of the definitions, the work supplies a useful reference list of formulations in the infinite-variable setting and confirms that the strong form carries over without additional restrictions. Such extensions are routine in commutative algebra but can serve as a consolidated resource when working with finitely supported polynomials or direct reductions to the finite case.

minor comments (3)

The introduction should explicitly list or tabulate the equivalent formulations (rather than only describing them as 'long') so that readers can quickly locate the precise statements being proved equivalent.
Clarify the precise meaning of 'large polynomial rings' in the persistence theorem (e.g., by citing the cardinality condition or the precise ring extension used) at the first appearance of the result.
Add a short remark on whether the equivalences require the base ring to be Noetherian or algebraically closed, or whether they hold more generally.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. The manuscript compiles equivalent formulations of Hilbert's Nullstellensatz in the infinite-variable setting and establishes persistence of the strong form in large polynomial rings. No specific major comments are listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper compiles a list of equivalent formulations of the Nullstellensatz in infinite dimensions and proves persistence of the strong form in large polynomial rings. No load-bearing steps reduce by construction to inputs, self-definitions, fitted parameters renamed as predictions, or self-citation chains. Equivalences are established via direct extension of standard definitions to finitely supported polynomials or reduction to the finite-variable case, which is self-contained mathematical verification without circular reduction. The derivation relies on classical commutative algebra notions that do not presuppose the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available. No free parameters, invented entities, or paper-specific axioms are mentioned. The work rests on whatever background commutative algebra is assumed in the full manuscript.

axioms (1)

standard math Standard definitions and properties of polynomial rings, ideals, and varieties in commutative algebra
The Nullstellensatz and its variants presuppose these background facts.

pith-pipeline@v0.9.0 · 5530 in / 1164 out tokens · 36812 ms · 2026-05-23T02:51:48.504253+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We compile a long list of equivalent formulations of Hilbert’s Nullstellensatz in infinite dimensions, and prove a persistence result for the strong Nullstellensatz in large polynomial rings.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Enrique Arrondo, Another elementary proof of the Nullstellensatz , Amer. Math. Monthly 113 (2006), no. 2, 169–171. DOI 10.1080/00029890.2006.11920292 . MR2203239

work page doi:10.1080/00029890.2006.11920292 2006
[2]

Markus Brodmann, Algebraische Geometrie, Birkh¨ auser Verlag, Basel, 1989

work page 1989
[3]

Schlank, and Alle Yuan, The Chromatic Nullstellensatz , Annals of Mathematics (to appear)

Robert Burklund, Tomer M. Schlank, and Alle Yuan, The Chromatic Nullstellensatz , Annals of Mathematics (to appear). Available at https://arxiv.org/abs/2207.09929

work page arXiv
[4]

Available at http://alpha.math.uga.edu/~ pete/integral2015.pdf

Pete L Clark, Commutative Algebra , (lecture notes), 2015. Available at http://alpha.math.uga.edu/~ pete/integral2015.pdf

work page 2015
[5]

Lev Glebsky and Carlos Jacob Rubio-Barrios, A simple proof of Hilbert’s Nullstellensatz based on Gr¨ obner bases, Lect. Mat. 34 (2013), no. 1, 77–82. [spanish version] . MR3085773

work page 2013
[6]

Oscar Goldman, Hilbert rings and the Hilbert Nullstellensatz , Math. Z. 54 (1951), 136–140. DOI 10.1007/BF01179855 . MR0044510

work page doi:10.1007/bf01179855 1951
[7]

David Hilbert, ¨Uber die Theorie der algebraischen Formen , Math. Ann. 36 (1890), no. 4, 473–534. Available at http://eudml.org/doc/157506

work page
[8]

MR0345945

Irving Kaplansky, Commutative rings , Revised edition, University of Chicago Press, Chicago, Ill.-London, 1974. MR0345945

work page 1974
[9]

W olfgang Krull, Jacobsonsche Ringe, Hilbertscher Nullstellensatz, Dimen sionstheorie, Math. Z. 54 (1951), 354–387. DOI 10.1007/BF01238035 . MR0047622

work page doi:10.1007/bf01238035 1951
[10]

Serge Lang, Hilbert’s Nullstellensatz in inﬁnite-dimensional space , Proc. Amer. Math. Soc. 3 (1952), 407–410. DOI 10.2307/2031893 . MR0047019

work page doi:10.2307/2031893 1952
[11]

Peter May, Munshi’s proof of the Nullstellensatz , Amer

J. Peter May, Munshi’s proof of the Nullstellensatz , Amer. Math. Monthly 110 (2003), no. 2, 133–140. DOI 10.2307/3647772 . MR1952440

work page doi:10.2307/3647772 2003
[12]

Oscar Zariski, A new proof of Hilbert’s Nullstellensatz , Bull. Amer. Math. Soc. 53 (1947), 362–368. DOI 10.1090/S0002-9904-1947-08801-7 . MR0020075 Mathematisches Institut, Universit ¨at T ¨ubingen, Auf der Morgenstelle 10, 72076 T¨ubingen, Germany Email address : zeidler@math.uni-tuebingen.de

work page doi:10.1090/s0002-9904-1947-08801-7 1947

[1] [1]

Enrique Arrondo, Another elementary proof of the Nullstellensatz , Amer. Math. Monthly 113 (2006), no. 2, 169–171. DOI 10.1080/00029890.2006.11920292 . MR2203239

work page doi:10.1080/00029890.2006.11920292 2006

[2] [2]

Markus Brodmann, Algebraische Geometrie, Birkh¨ auser Verlag, Basel, 1989

work page 1989

[3] [3]

Schlank, and Alle Yuan, The Chromatic Nullstellensatz , Annals of Mathematics (to appear)

Robert Burklund, Tomer M. Schlank, and Alle Yuan, The Chromatic Nullstellensatz , Annals of Mathematics (to appear). Available at https://arxiv.org/abs/2207.09929

work page arXiv

[4] [4]

Available at http://alpha.math.uga.edu/~ pete/integral2015.pdf

Pete L Clark, Commutative Algebra , (lecture notes), 2015. Available at http://alpha.math.uga.edu/~ pete/integral2015.pdf

work page 2015

[5] [5]

Lev Glebsky and Carlos Jacob Rubio-Barrios, A simple proof of Hilbert’s Nullstellensatz based on Gr¨ obner bases, Lect. Mat. 34 (2013), no. 1, 77–82. [spanish version] . MR3085773

work page 2013

[6] [6]

Oscar Goldman, Hilbert rings and the Hilbert Nullstellensatz , Math. Z. 54 (1951), 136–140. DOI 10.1007/BF01179855 . MR0044510

work page doi:10.1007/bf01179855 1951

[7] [7]

David Hilbert, ¨Uber die Theorie der algebraischen Formen , Math. Ann. 36 (1890), no. 4, 473–534. Available at http://eudml.org/doc/157506

work page

[8] [8]

MR0345945

Irving Kaplansky, Commutative rings , Revised edition, University of Chicago Press, Chicago, Ill.-London, 1974. MR0345945

work page 1974

[9] [9]

W olfgang Krull, Jacobsonsche Ringe, Hilbertscher Nullstellensatz, Dimen sionstheorie, Math. Z. 54 (1951), 354–387. DOI 10.1007/BF01238035 . MR0047622

work page doi:10.1007/bf01238035 1951

[10] [10]

Serge Lang, Hilbert’s Nullstellensatz in inﬁnite-dimensional space , Proc. Amer. Math. Soc. 3 (1952), 407–410. DOI 10.2307/2031893 . MR0047019

work page doi:10.2307/2031893 1952

[11] [11]

Peter May, Munshi’s proof of the Nullstellensatz , Amer

J. Peter May, Munshi’s proof of the Nullstellensatz , Amer. Math. Monthly 110 (2003), no. 2, 133–140. DOI 10.2307/3647772 . MR1952440

work page doi:10.2307/3647772 2003

[12] [12]

Oscar Zariski, A new proof of Hilbert’s Nullstellensatz , Bull. Amer. Math. Soc. 53 (1947), 362–368. DOI 10.1090/S0002-9904-1947-08801-7 . MR0020075 Mathematisches Institut, Universit ¨at T ¨ubingen, Auf der Morgenstelle 10, 72076 T¨ubingen, Germany Email address : zeidler@math.uni-tuebingen.de

work page doi:10.1090/s0002-9904-1947-08801-7 1947