Hilbert's Tenth Problem over Rings

• Hilbert's Tenth Problem over rings is defined by examining the algorithmic undecidability of solving polynomial equations in complementary subrings of number fields.
• The approach uses a rank one elliptic curve to construct Diophantine models that simulate integer arithmetic via x-coordinate divisibility and primitive prime ideal factors.
• The emergence of infinite discrete Diophantine sets challenges Mazur’s local-global conjectures, highlighting profound implications for computational number theory and arithmetic geometry.

Hilbert’s Tenth Problem over rings—especially over complementary subrings of number fields—concerns the existence of general algorithms to determine solvability of polynomial equations in these algebraic structures. The paper of its undecidability not only extends the original result for the integers to other domains but also reveals deep connections between Diophantine definability, arithmetic of elliptic curves, construction of Diophantine sets, and the topology of number fields. The interplay with Mazur’s conjectures and the precision with which subrings can be constructed further highlights the subtleties introduced by the structure of number fields and their primes.

1. Complementary Subrings and Hilbert’s Tenth Problem

Let $K$ be a number field and $\mathcal{O}_K$ its ring of integers. For a subset $S$ of the set of nonarchimedean primes (finite primes) of $K$, the big ring $\mathcal{O}_{K,S}$ consists of all $x \in K$ integral outside $S$: $$\mathcal{O}_{K,S} = \{ x \in K \mid \operatorname{ord}_\mathfrak{p}(x) \ge 0 \text{ for all } \mathfrak{p} \notin S \}.$$ A complementary subring is obtained by partitioning the set of nonarchimedean primes (possibly with restrictions on density) into disjoint recursive subsets $S_1, \ldots, S_t$ and considering the associated subrings $\mathcal{O}_{K,S_i}$ with $S_i$ the set of primes inverted in the construction of the localized ring. The primary question is whether Hilbert's Tenth Problem (HTP) is undecidable in such subrings; i.e., whether there exists no algorithm capable of determining the solvability of general Diophantine equations in $\mathcal{O}_{K,S}$.

2. Diophantine Models and Simulation of Integers

The undecidability of HTP in $\mathcal{O}_{K,S}$ is established via the construction of a Diophantine model of $\mathbb{Z}$ inside $\mathcal{O}_{K,S}$. For an elliptic curve $E/K$ of rank one, there exists a $K$-rational point $P$ of infinite order. The sequence $(nP)_{n \in \mathbb{Z}}$ gives rise to x-coordinates $x(nP)$ whose denominators—when factored in $\mathcal{O}_K$—encode intricate divisibility properties. The core construction analytically proceeds as follows:

• Each $x(nP)$ can be written in projective coordinates as $(C_n,U_n)$, and the associated divisor satisfies $(x(nP)) = a_n/b_n$. Primitive divisors of $b_n$ for large $n$ provide new prime ideals.
• By judiciously selecting a sequence of integers $(\ell_i)$ and examining $A = \{ x(\ell_iP) \mid i \in \mathbb{Z}_+ \}$, one builds an infinite Diophantine set $A \subset \mathcal{O}_{K,S}$.
• The addition and (via further relations) multiplication structure of $\mathbb{Z}$ can be Diophantinely encoded among the $x(\ell_iP)$ by controlling divisibility at selected primes $\mathfrak{p}$: equations of the form

$$\operatorname{ord}_\mathfrak{p}\bigl(x(\ell_iP) - x_1\bigr) = c + i$$

for fixed $x_1$ and $c$ simulate the successor relation. The full graph of addition is then defined Diophantinely in this framework.

Through this simulation, one obtains a Diophantine correspondence $o: \mathbb{Z}_+ \to A$, and since HTP is undecidable for $\mathbb{Z}$, undecidability follows for $\mathcal{O}_{K,S}$.

3. Recursive Partitioning of Primes with Prescribed Density

A distinctive innovation is the partitioning of all nonarchimedean primes of $K$ into $t$ disjoint recursive subsets $S_1,\ldots,S_t$ with specified computable natural densities $(\delta_1,\ldots,\delta_t)$ satisfying $\delta_1 + \dots + \delta_t = 1$. Given such densities (which can be arbitrary computable nonnegative real numbers summing to one), the construction proceeds by:

• Starting with recursive sets and modifying them via the addition or removal of auxiliary recursively enumerable sets (such as $T_{1,r}$, $T_{2,r}$), which control the appearance of "marked" primes as primitive ideal divisors of $x(nP)$ sequences.
• The auxiliary sets themselves have density zero, but their alteration assures that the final sets $S_i$ meet both the density and the recursive requirements.

This technical partitioning (see Proposition 8.9 in the original work) permits covering the spectrum of big rings $\mathcal{O}_{K,S_i}$ with precise density control, greatly extending the granularity with which one may select the primes inverted in the big rings.

4. Infinite Discrete Diophantine Sets and Mazur's Conjectures

A crucial feature of the construction is the existence, in each $\mathcal{O}_{K,S_i}$, of an infinite Diophantine set that is discrete in every topology on $K$—both archimedean and nonarchimedean. The set

$A = \{ x(\ell_iP) \mid i \in \mathbb{Z}_+ \}$

is Diophantine and, per choice of the sequence $\ell_i$, ensures discreteness: for any place $v$ of $K$, the points $x(\ell_iP)$ accumulate only at a designated limit outside $A$ in $K_v$, and are thus isolated within $A$. These infinite discrete sets deliver counterexamples to the local versions of Mazur’s conjecture on the topology of rational points, which would otherwise predict that no infinite Diophantine set can be discrete in the $p$-adic or real topology. The failure of these conjectures in big rings signifies a fundamental divergence in Diophantine-topological behavior between $\mathcal{O}_K$ and its localizations at large or thin sets of primes.

5. Role and Necessity of Elliptic Curves of Rank One

The foundation of the undecidability mechanism is the existence of an elliptic curve $E/K$ with rank one. The consequences are multifold:

• Existence of a $K$-rational point of infinite order is required for generating the necessary divisibility sequences with many primitive divisors.
• The structure of the Mordell–Weil group $E(K)$, equipped with the ability to transfer properties through localizations to various completions $K_v$ via isomorphisms to $(\mathbb{R}/\mathbb{Z})^n$ at archimedean places, guarantees the convergence and isolation properties needed for discreteness.
• Without a rank one elliptic curve, the method for producing these Diophantine sets and controlling the distribution of prime divisors breaks down; the construction of the necessary Diophantine model becomes unfeasible.

6. Interaction with Mazur’s Local-Global Principles and Broader Impact

The demonstrated undecidability in $\mathcal{O}_{K,S}$ shows that local-global principles known or conjectured for the full ring of integers can fail spectacularly in big rings. Specifically:

• Mazur's conjecture on the topological closure of rational points (predicting no infinite discrete Diophantine sets in the archimedean or nonarchimedean topology) is invalidated for complementary subrings, despite holding (or expected to hold) for $\mathcal{O}_K$.
• The introduction of density partitions and the construction of discrete sets elucidate the limitations of transferring topological conjectures from the global to large subring settings.

Moreover, these developments highlight the robustness of undecidability for HTP under local modifications of the ring—undecidability persists not only in $\mathcal{O}_K$ but also in a wide array of its big subrings, even as the set of inverted primes is tailored to arithmetic or density constraints.

Table: Key Mathematical Constructs in the Approach

Construct	Role in Undecidability	Source/Formula
Elliptic curve $E/K$ of rank one	Generates divisibility sequences	Multiples $nP$, factorization $(x(nP)) = a_n/b_n$
Diophantine set $A$ via $x(\ell_iP)$	Diophantine model/discrete set in ring	$A = \{ x(\ell_iP) \mid i \in \mathbb{Z}_+ \}$
Partition of primes into $S_1,\ldots,S_t$	Construction of big rings with density	Recursive sets, densities $\delta_i$ ($\sum \delta_i = 1$)
Primitive ideal divisors	Mark primes for Diophantine operations	Via divisibility in denominators of $x(nP)$
Discreteness in all topologies	Violates Mazur’s local-global conjectures	Sequencing to control accumulation in completions $K_v$
Diophantine simulation of $\mathbb{Z}$	Reduces undecidability to known case	Relation: $$\operatorname{ord}_\mathfrak{p}(x(\ell_iP) - x_1) = c + i$$

7. Conclusion

Hilbert’s Tenth Problem is undecidable in every complementary subring $\mathcal{O}_{K,S}$ derived via recursive partitions with prescribed density, provided $K$ admits an elliptic curve of rank one (Eisentraeger et al., 2010). The technical apparatus—leveraging rank-one elliptic curves, primitive ideal divisors, explicit density partitioning, and the design of infinite discrete Diophantine sets—shows both the depth and nuance in transferring undecidability phenomena from $\mathbb{Z}$ to broad classes of number-theoretic rings. These results not only provide broad generalizations of the original negative solution to HTP but also illuminate limitations on topological conjectures such as those of Mazur, enhance our understanding of Diophantine logic over big rings, and define precise mathematical frameworks for further exploration of Diophantine undecidability in arithmetic geometry.