Undecidability in the Ramsey theory of polynomial equations and Hilbert's tenth problem

Bill Mance; Sohail Farhangi; Steve Jackson

REVIEW 3 major objections 4 minor 2 cited by

Undecidability in the Ramsey theory of polynomial equations and Hilbert's tenth problem

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Over $F_q(t)$, no algorithm decides which polynomial equations have monochromatic solutions.

desk verdict Strong paper with a real gap: the Pi-2-completeness results are conditional on master polynomials whose existence is asserted but not proved for the function-field cases. read the letter →

arxiv 2412.14917 v2 pith:KZ46C6B3 submitted 2024-12-19 math.LO math.COmath.DS

Sohail Farhangi , Steve Jackson , Bill Mance This is my paper · ORCID

classification math.LOmath.COmath.DS MSC 03D8005D1011U05

keywords partitionregularitydensityRamseytheoryhomogeneouspolynomialequationsarithmeticalhierarchyPi-0-2completenessmasterpolynomialsamenablesemigroupsundecidability

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks how hard it is to decide whether a given polynomial equation has a monochromatic solution in every finite coloring of a number system's nonzero elements. It proves that for many integral domains, including rational function fields such as $F_q(t)$, this decision problem is not merely undecidable but sits exactly at the $\Pi^0_2$ level of the arithmetical hierarchy, and that the same holds for the injective and density-regular variants. Over the integers the result is conditional on the undecidability of the rational analogue of the classic integer-root problem, which is still open. The proof works by encoding arbitrary $\Pi^0_2$ questions as the partition regularity or density regularity of homogeneous polynomials, using what the paper calls master polynomials.

What carries the argument

The engine is the master polynomial. For a domain $R$ with fraction field $K$, a master polynomial for a $\Sigma^0_1$ set $S\subseteq\mathbb{N}\times\mathbb{Z}$ (or $\mathbb{N}\times\mathbb{Q}$) is a fixed polynomial $p(x,y,z_1,\dots,z_k)$ together with computable maps $f:\mathbb{N}\to R$ and $g:K^\times\to\mathbb{Z}$ (or $g:K^\times\to\mathbb{Q}^\times$) such that $S(m,n)$ holds exactly when $p(f(m),b,z_1,\dots,z_k)$ has a root in $K^\times$ for every $b$ with $g(b)=n$, and fails exactly when it has no root in $K$ for every such $b$. The paper converts these into homogeneous polynomials by substituting scaled quotients such as $(z_i-z_j)/(z_k-z_\ell)$ or $(z_i-z_j)/z_k$, so that a root of the master polynomial becomes an injective root of a homogeneous polynomial. Two self-contained tools carry the density arguments: a compactness principle saying that density regularity of a right-translation-invariant family of finite configurations is equivalent to a statement about a single finite set, and a uniformity principle producing a finite subfamily with a uniform positive recurrence bound; both are proved for countable cancellative left amenable semigroups.

What would settle it

Take the standard 'for all $n$ there exists $m$' complete problem and run the paper's reduction over $F_q(t)$: it outputs a homogeneous polynomial $p_m$ for each index $m$. If one could exhibit an $m$ where $p_m=0$ is partition regular but the universal statement is false, or $p_m=0$ is not partition regular but the universal statement is true, the reduction would be refuted; the theorem asserts no such $m$ exists.

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Extended reading notes

Core claim

The paper's central result is a complexity classification. If a computable integral domain $R$ admits master polynomials, then the set of homogeneous polynomials $p$ for which $p=0$ is partition regular over $R\setminus\{0\}$ is $\Pi^0_2$-complete, and the same is true for the injective partition-regular version and for the multiplicative density-regular analogue $\mathrm{IMDR}_R\cap H_R$. When the number of cells in the partition is fixed at $\ell$, the corresponding sets are $\Sigma^0_1$-complete rather than $\Pi^0_2$-complete, assuming the undecidability of the root-existence problem over the fraction field. An unconditional special case is $R=F_q[t]$ with fraction field $F_q(t)$: deciding whether a homogeneous polynomial is partition regular over $F_q(t)\setminus\{0\}$ is $\Pi^0_2$-complete, hence undecidable. For $R=\mathbb{Z}$, the same conclusion is conditional on the open problem of deciding whether polynomials over $\mathbb{Q}$ have rational roots. Along the way the paper proves compactness and uniformity principles for density Ramsey theory on countable cancellative left amenable semigroups, and constructs natural extensions of measure-preserving semigroup actions for countable cancellative left reversible semigroups.

Load-bearing premise

The load-bearing premise is that the ring admits master polynomials, which for $\mathbb{Z}$ is equivalent to the open question of whether rational-root existence is undecidable, and for the function fields is inherited from known undecidability results.

Editorial extensions

If this is right

For every domain covered by the theorem, the set of homogeneous polynomials $p$ for which $p=0$ is partition regular over $R\setminus\{0\}$ is $\Pi^0_2$-complete, so no algorithm can decide it.
The injective version, where the monochromatic solution must use distinct variables, has the same $\Pi^0_2$-complete complexity, and the multiplicative density-regular analogue does too.
Fixing the number of colors $\ell$ changes the complexity: the $\ell$-partition-regular sets are $\Sigma^0_1$-complete, so the quantifier over all $\ell$ is what raises the problem to $\Pi^0_2$.
For $R=\mathbb{Z}$, deciding partition regularity of homogeneous polynomials would be $\Pi^0_2$-complete if the open question of rational-root existence over $\mathbb{Q}$ has a negative answer; the function-field cases give unconditional instances.
The compactness and uniformity principles imply that density regularity has finite witnesses: if every set of upper Banach density at least $\delta$ contains a configuration from a right-translation-invariant family, then a single finite subfamily works uniformly for all such sets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not claim that every domain with an undecidable root-existence problem over its fraction field yields $\Pi^0_2$-completeness, but the master-polynomial mechanism suggests the conclusion should extend to any such domain once a suitable valuation homomorphism is available.
Beyond the paper's scope, a settled answer to the rational-root question would automatically resolve the integer case, so the polynomial partition-regularity problem can be read as a reformulation of that open problem.
An unstated practical upshot of the uniformity principle is that verifying density regularity of a specific polynomial could be reduced to a finite search over one large finite set, turning a $\Pi^0_2$ condition into an explicit finite certificate check.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read this one because it connects two big programs: Hilbert's tenth problem over Q and the descriptive complexity of partition- and density-regular polynomial equations. The main claims are that the sets PR, IPR, IMDR are Pi_2^0-complete over suitable domains, with Z conditional on HTP(Q). The reductions themselves are clever and, as far as I can see, correct: they realize ratios inside dense or partition-regular sets and use the master polynomial as a bridge. The compactness and uniformity principles in Section 3 look solid and are independently useful.

But the load-bearing assumption is Definition 2.26, the 'master polynomial,' and Lemma 2.28 that supposedly guarantees its existence. This is the weak spot. For Z, the proof says 'let f and g be the identity functions.' In the additive formulation that is type-incorrect—g would need to map Q^x onto Z, and identity maps to Q^x. The multiplicative version with g=id is fine for the cases where a single parameter n in Q is used, and HTP(Q) does give that. So the Z part may survive with a rewrite.

The bigger problem is the function-field case. Lemma 2.28(2) cites Pheidas, Videla, Shlapentokh for the existence of master polynomials with g=ord_p, but gives no theorem numbers and no construction. The cited results are about Diophantine models of N over F_q(t), not about the universal-fiber condition in Definition 2.26. That condition requires that for every n in the image of g, the Diophantine set {b in K : g(b)=n, p(m,b,z)=0 has a root} is actually constant on the whole fiber. Standard Diophantine models give you one parameter t^n per n, not uniformity across the infinitely many unit multiples. I don't see how the citations as written deliver that. If this cannot be proved, the Pi_2^0-completeness for IMDR and PR over F_q(t) does not go through.

I want to be clear: this is not a fatal disorganization. The motivation is good, the paper is honest about what is conditional, and the reductions between the Ramsey-theoretic sets and HTP are the opposite of circular—they reduce HTP to the Ramsey sets. But the master polynomial lemma is underproved, and it is central.

Verdict: send it to a serious referee. The referee should ask for a full proof or a precise reference of the master polynomial construction in the function-field setting. If the authors can supply that, this becomes a very nice paper. If not, the completeness results collapse to conditional statements. I would not cite the main theorem as it stands, though the compactness principle in Section 3 is worth a look.

Referee Report

3 major / 4 minor

Summary. The paper studies the lightface descriptive complexity of Ramsey-theoretic sets of polynomials over computable integral domains R: partition-regular polynomials PR_R, injectively partition-regular IPR_R, and injective multiplicative/additive density-regular polynomials IMDR_R and IADR_R. The main completeness result is Theorem 4.2: if R admits master polynomials (Definition 2.26), then PR_R, IPR_R, PR_R∩H_R, IPR_R∩H_R, and IMDR_R∩H_R are Π_2^0-complete. Theorem 4.1 gives Σ_1^0-completeness of fixed-color and fixed-density versions under the undecidability of HTP(K). The paper also proves compactness and uniformity principles for density Ramsey theory on countable cancellative left amenable semigroups and constructs natural extensions of measure-preserving actions. The proofs are built from explicit transfer lemmas (Lemma 2.15, Corollary 2.21, Lemma 2.23), Rado's theorem over integral domains, and tiling arguments. Unconditional conclusions are claimed for F_q[t] and rings of integers of algebraic function fields over finite fields, and the Z case is conditional on HTP(Q).

Significance. If the master-polynomial hypothesis is fully established, the exact Π_2^0-completeness results are substantial and novel: they give a precise sense in which deciding partition regularity and density regularity of polynomial equations is as hard as the universal fragment of arithmetic. The compactness principle (Theorem 3.1), uniformity principle (Theorem 3.7), and natural extension construction (Theorem 2.9) are useful contributions in their own right, and the paper is explicit about which claims are conditional, including open questions and concrete falsifiable predictions such as PR_{F_q[t]} being Π_2^0-complete. The main weakness is that the upgrade from Σ_1^0-hardness to Π_2^0-completeness rests on the unproved existence of master polynomials in the strong sense of Definition 2.26; the current verification in Lemma 2.28 is too terse and appears type-incorrect in the additive case.

major comments (3)

[§2.7, Lemma 2.28(1)] The proof says 'we let f and g be the identity functions' and concludes that Z admits master polynomials. This is type-incorrect for the additive case of Definition 2.26: g must be a surjective homomorphism from Q^x to Z, so g = id is impossible. The argument at best establishes the multiplicative case, where g: Q^x → Q^x can be the identity and HTP(Q) gives the required reduction. Since Theorem 4.2 invokes master polynomials without specifying which type is used, the conditional result for R = Z is not proved as written.
[§2.7, Lemma 2.28(2)] For function fields, the proof consists of citing [50, 62, 57] and asserting that these results 'construct master polynomials' with g = ord_p. Definition 2.26 requires much more than ordinary Diophantine undecidability: for each fiber of g, either every b in the fiber admits a root in K^x (if S holds) or no b in the fiber admits a root in K at all (if S fails). This universal-fiber property, together with the negative 'no root in K' condition, is not the standard formulation of HTP(K), and no construction or theorem number is supplied. The unconditional Π_2^0-completeness claims in Theorem 4.2 are therefore not supported as they stand.
[§4, Theorem 4.3(i)] In the converse direction, the proof chooses a monotile T for (Q,+) with center set C, asserts d*(C) = 1/|T|, and uses this to produce a positive-density set avoiding the forbidden differences. But IADR_Z requires a subset of Z, not of Q. For a general center set C of an additive tiling of Q, the intersection C∩Z need not have positive density in Z; the argument needs an explicit construction (for example, a coset M·Z chosen to avoid the finite set T) with a proof that the resulting set lies in Z and has positive density.

minor comments (4)

[§2.7, Definition 2.26] Footnote 7 says that a universal Σ_1^0 set S ⊆ N×Q admits 'an additive master polynomial'; this should read 'a multiplicative master polynomial'.
[§4, Theorem 4.2] In the proof of part (i), the text says 'M P2(m, y1, y2, z1, ..., z4k)' in the converse directions; the intended polynomial is M P1. The same typo appears in the multiplicative case.
[§2.7, Lemma 2.28(1)] The displayed verification 'for all m, n ∈ N' is appropriate for the additive formulation but not for the multiplicative master polynomial; in the multiplicative case n should range over Q.
[§2.7] The transformation from a Π_2^0 set A to a Σ_1^0 set S with A(m) iff ∀n≠0 S(m,n) is stated correctly but deserves a short explicit proof; the current text compresses the quantifier manipulation and could confuse readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper reduces HTP(K) to Ramsey-theoretic sets, and the two self-citations are independent prior results rather than presuppositions of the target claims.

full rationale

The main derivations run from Hilbert's tenth problem (or the assumed existence of master polynomials) to the Π_2^0-completeness of partition-regular and density-regular polynomial sets. This is a reduction of an external undecidable problem to the Ramsey-theoretic sets, not a reduction of the Ramsey sets to themselves. The master polynomial hypothesis in Definition 2.26 is an imported assumption, not a conclusion derived from the target sets, so no 'fitted input called prediction' pattern appears. The one self-citation, Lemma 2.23, is drawn from the independently published Theorem 27 of Farhangi and Magner [29] and concerns partition regularity of specific ratio equations; it is used as a tool inside the proof of Theorem 4.2, and its statement is external to the paper's own completeness claims. It is load-bearing, but it is not unverified self-support in the circular sense: the cited result has its own proof elsewhere and is not equivalent to the paper's conclusions. The proof of Lemma 2.28 is brief and, for the additive case with g = identity, appears type-incorrect because the identity map from Q^x to Z is not a homomorphism with the stated codomain. However, that is a correctness or verification concern about an imported hypothesis, not a circularity: the paper does not claim to derive master polynomials from the Ramsey sets. The reductions in Theorems 4.1 and 4.2 are one-directional and give new content (exact lightface complexity) beyond the HTP input. Hence the derivation chain does not collapse to its own assumptions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

There are no fitted parameters or invented physical entities. The main load-bearing external inputs are the master polynomial hypothesis, HTP(Q) for the integer case, Lemma 2.23 from the first author's earlier work, and standard Ramsey/ergodic theorems. The master polynomials are introduced in this paper as a definition, but their existence is imported from prior HTP literature.

assumptions (5)

domain assumption Hilbert's tenth problem over Q is undecidable (assumed for the R = Z case).
Open problem; Theorem 4.2(ii) and Theorem 4.3(i) are explicitly conditional on it, and Lemma 2.28(1) shows Z admits master polynomials only under this assumption.
ad hoc to paper R admits master polynomials in the sense of Definition 2.26.
This strong uniform Diophantine definability is not established in the paper for general R; it is imported from known HTP results for function fields and from HTP(Q) for Z. The Pi_2^0-completeness theorem is stated for exactly this class.
domain assumption Lemma 2.23 (drawn from Farhangi and Magner, Integers 2023, Theorem 27).
Used in the forward direction of Theorem 4.2 and in Lemma 4.5; its proof is not reproduced here and it is a nontrivial ultrafilter statement about partition regularity of ratio equations.
standard math Standard Ramsey and ergodic theorems: Rado's theorem for integral domains, Furstenberg correspondence, Lindenstrauss pointwise ergodic theorem, Austin's theorem, and monotileability of countable abelian groups.
These are cited external results used in Section 3 and Corollary 2.21; they are canonical and not proved in full.
standard math Ore embedding theorem: cancellative left reversible semigroups embed in their group of right quotients.
Used throughout Section 3 to pass from semigroups to amenable groups; stated as Theorem 2.5 with references.

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Cite this review

Pith. "Pith review of Undecidability in the Ramsey theory of polynomial equations and Hilbert's tenth problem." pith.science (2026). https://pith.science/paper/KZ46C6B3

@misc{pith2026241214917,
  author       = {Pith},
  title        = {Pith review of: Undecidability in the Ramsey theory of polynomial equations and Hilbert's tenth problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZ46C6B3}},
  note         = {Machine review of arXiv:2412.14917}
}

abstract

We show that several sets of interest arising from the study of partition regularity and density Ramsey theory of polynomial equations over integral domains are undecidable. In particular, we show that the set of homogeneous polynomials $p \in \mathbb{Z}[x_1,\cdots,x_n]$ for which the equation $p(x_1,\cdots,x_n) = 0$ is partition regular over $\mathbb{Z}\setminus\{0\}$ is undecidable conditional on Hilbert's tenth problem for $\mathbb{Q}$. For other integral domains, we get the analogous result unconditionally. More generally, we determine the exact lightface complexity of the various sets of interest. For example, we show that the set of homogeneous polynomials $p \in \mathbb{F}_q(t)[x_1,\cdots,x_n]$ for which the equation $p(x_1,\cdots,x_n) = 0$ is partition regular over $\mathbb{F}_q(t)\setminus\{0\}$ is $\Pi_2^0$-complete. We also prove several other results of independent interest. These include a compactness principle and a uniformity principle for density Ramsey theory on countable cancellative left amenable semigroups, as well as the existence of the natural extension for measure preserving systems of countable cancellative left reversible semigroups.

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