Hilbert’s tenth problem over the rationals: a claimed limit of algorithms

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Family 004 · Number theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

Suppose a computer receives a polynomial with integer coefficients, such as

$$f(X,Y)=X^2+Y^2-3.$$

Can one algorithm always decide whether the equation $f=0$ has a solution whose coordinates are rational numbers? The algorithm may take a long time, but it must eventually return the correct yes or no for every input. Both the polynomial and its number of variables are allowed to vary.

This is Hilbert’s tenth problem over $\mathbb Q$. Family 004’s main manuscript, M1, claims that no such algorithm exists. It further places the problem at the same level of Turing unsolvability as the halting problem and retains that conclusion for polynomials of degree at most four.

A companion, M2, supplies an elliptic-curve theorem used in the construction. The connection is unexpected: information about rational points and L-functions becomes part of an argument about the limits of computation. The route does not produce a single polynomial definition of the integers inside the rationals, a distinction essential to understanding the claim.

2. Background and essential definitions

A polynomial is a finite sum of monomials with specified coefficients. A rational zero of $f\in\mathbb Z[X_1,\ldots,X_n]$ is a tuple $(a_1,\ldots,a_n)\in\mathbb Q^n$ for which $f(a_1,\ldots,a_n)=0$. Rational and integer solvability differ: $2X-1=0$ has a rational solution but no integer solution. Allowing fractions does not preserve an integer-solvability problem unchanged.

A decision algorithm must halt on both yes and no instances. Rational solvability is already computably enumerable: enumerate all rational tuples and test them exactly. If a solution exists, this search eventually finds it. If none exists, enumeration continues forever. The missing ingredient is a procedure that also settles every negative instance.

The halting problem asks whether a given program eventually stops on a given input. A Turing reduction to another problem is an algorithm allowed to ask an oracle for answers to that other problem, possibly several times and adaptively. Two problems have the same Turing degree if each can be solved with an oracle for the other. The degree of the halting problem is denoted $0'$.

An existential polynomial condition asks whether additional rational witnesses exist satisfying specified polynomial equations. Saying that $\mathbb Z$ is Diophantine in $\mathbb Q$ would mean that there is one fixed polynomial $P$ such that

$$x\in\mathbb Z \quad\Longleftrightarrow\quad \exists y_1,\ldots,y_m\in\mathbb Q: P(x,y_1,\ldots,y_m)=0.$$

That would provide a direct translation of integer problems. A Turing reduction need not have this form.

The arithmetic ingredients include elliptic curves: smooth cubic curves with a distinguished identity point and an addition law on their rational points. If an infinite-order point $P$ generates a rank-one subgroup, its multiples $nP$ can encode integers $n$. The challenge is to recover multiplication of those indices using rational polynomial conditions; the curve’s addition law directly provides only addition.

3. Development history and previous partial results

Hilbert’s original 1900 problem concerned integer solutions. Davis, Putnam, and Robinson connected equations to computably enumerable sets, and Matiyasevich’s 1970 work completed the polynomial representation theorem. Every computably enumerable relation on positive integers can be encoded by polynomial solvability. As a consequence, integer solvability cannot be decided by an algorithm. The 1961 Davis–Putnam–Robinson paper is a principal milestone in that development.

Transferring this conclusion to rational solutions has been much harder. Julia Robinson’s 1949 work defined integers within rational arithmetic using both existential and universal quantifiers. Later refinements culminated in Koenigsmann’s universal definition. Universal quantifiers mean “for every rational number”; polynomial solvability permits only existential witnesses. The quantifier difference prevents the immediate transfer one might expect.

Another approach enlarges the integers by allowing denominators from a chosen set of primes $S$. The resulting ring is $\mathbb Z[S^{-1}]$. Poonen’s 2003 construction proves undecidability for certain such rings even when the permitted primes have density one. These remain proper subrings of $\mathbb Q$: a sparse excluded set can still carry crucial information.

More recently, Koymans and Pagano proved undecidability for every infinite ring finitely generated over $\mathbb Z$. Their work uses elliptic curves and additive combinatorics. The rational field is not finitely generated as a $\mathbb Z$-algebra: finitely many generators cannot introduce denominators involving every prime. Thus this major result does not itself settle the rational problem.

Geometry also cautions against identifying undecidability with an existential definition of the integers. Mazur’s conjecture about closures of rational points would obstruct such a definition and even certain Diophantine models of integer arithmetic. M1 pursues a finite-test oracle reduction instead. Its conclusion is designed to address the decision problem without asserting one of these stronger definability statements.

4. What the new manuscripts claim

Rational solvability and its degree

M1, Theorem 1.1, states that no algorithm decides, for an arbitrary integer-coefficient polynomial in an arbitrary finite number of variables, whether it has a rational zero.

Corollary 7.1 strengthens the computational description: the set of solvable inputs has Turing degree $0'$. The same degree is asserted for each of these restricted input forms:

• a polynomial of total degree at most four;
• a supplied finite nonempty list of polynomials $q_1,\ldots,q_s$, each of degree at most two, asking whether $\sum_{i=1}^s q_i^2$ has a rational zero.

There is no bound on the number of variables, the number of summands, or the sizes of the integer coefficients. The list is part of the second problem’s input; one need not recognize whether an arbitrary polynomial admits such a presentation.

The reduction from integer solvability is a Turing reduction, not a claimed single-instance many-one reduction. M1 explicitly does not establish many-one completeness or a single existential definition of $\mathbb Z$ in $\mathbb Q$.

The companion elliptic-curve statement

For an elliptic curve $E/\mathbb Q$, let $E(\mathbb Q)[2]$ be the subgroup of rational points killed by doubling. Let

$$s_2(E)=\operatorname{corank}_{\mathbb Z_2} \operatorname{Sel}_{2^\infty}(E/\mathbb Q).$$

The full 2-power Selmer group collects compatible arithmetic information from doubling and from all completions of $\mathbb Q$. Its corank counts its unbounded 2-divisible directions. It includes the rational-point rank and potentially a contribution from the Shafarevich–Tate group $\operatorname{Sha}(E/\mathbb Q)$, which measures certain failures to pass from local solutions to global ones.

M2, Theorem 1.1, assumes

$$E(\mathbb Q)[2]\ne0, \qquad s_2(E)\in\{0,1\}.$$

It concludes that the order of vanishing of $L(E,s)$ at $s=1$, the rational-point rank, and $s_2(E)$ are equal, and that the whole group $\operatorname{Sha}(E/\mathbb Q)$ is finite. No restriction on reduction at the prime two is imposed. A rank bound alone is not its hypothesis, and the theorem does not assert the complete BSD leading-coefficient formula.

M1 uses M2 together with the prime-two Fontaine–Mazur modularity claim discussed in family 010. M2 presents its own treatment of rational two-torsion, without using the broader converse as an input.

5. Why the advance matters

Provisional importance: 5/5. The rational version of Hilbert’s tenth problem is a central question connecting arithmetic geometry and logic. A negative answer would identify a fundamental limit on rational-point algorithms, not merely show that some existing method is too slow.

The quartic consequence is especially instructive. Bounded polynomial degree does not make the problem decidable when the number of unknowns is unrestricted. Complicated computation can be distributed across many simple equations and then assembled into one equation.

The distinction between a decision theorem and a definability theorem also matters conceptually. It would show that rational arithmetic can support maximal computably enumerable Turing complexity even through a reduction weaker than a direct existential encoding of all integer arithmetic. This rating is conditional on the manuscripts’ mathematical claims.

6. The main idea

Replace one translation by a sequence of tests

For an integer polynomial $f$, M1 constructs an effectively generated sequence of finite tests, each answerable by finitely many rational-solvability queries. Its key equivalence is

$$f\text{ has an integer zero} \quad\Longleftrightarrow\quad \text{every finite test succeeds}.$$

Assume temporarily that a rational-solvability algorithm exists. Run two searches together. One enumerates integer tuples looking for a zero. The other examines the tests looking for a failure. If the equivalence holds, exactly one search eventually certifies the correct answer. That would decide integer solvability, contradicting the classical theorem.

All the difficulty is therefore concentrated in showing that an integer-unsolvable equation eventually fails a test. A finite failure is the stopping certificate; no advance bound on which test will fail is required.

Force modeled indices to become ordinary integers

Suppose every test succeeds. The compactness theorem of logic says that a collection of first-order conditions has a model if every finite subcollection does. M1 uses this to obtain an ambient extension containing modeled root coordinates and elliptic points attached to ring elements.

These points have indices in an enlarged integer structure that may include numbers larger than every ordinary integer. The construction initially makes the index assignment additive and injective. It does not automatically preserve multiplication, so substituting these indices into $f$ need not yet give zero.

The manuscript compares the ring values and their elliptic indices prime by prime. For a rational number, the valuation $v_p$ records the exponent of $p$ in its numerator minus that in its denominator. Local elliptic logarithms turn the curve’s addition law into addition near its identity. Two conjugate local tests are used to recover index congruences while controlling denominators.

A global height estimate then joins these local comparisons. For coprime integers $u,v$, the logarithmic height of $(u:v)$ is

$$h(u:v)=\log\max\{|u|,|v|\}.$$

M1 selects five points on the projective line and looks at primes where a rational point meets one of them with odd valuation, outside a fixed exceptional set. If $M(s)$ is the product of those distinct primes, the claimed estimate has the form

$$h(s)\le C\,M(s)^c$$

for fixed positive constants $C,c$. This is an estimate for the logarithmic height; replacing it by a bound for $e^{h(s)}$ would change the assertion.

Write $\delta$ for $f$ evaluated at the modeled integer indices. If $\delta\ne0$, the local tests force the relevant contact primes to divide this same integer to high precision. The height estimate then bounds heights throughout the modeled ring in terms of the largest root index $B$. But the height of an elliptic multiple grows quadratically in its index. The resulting comparison has the shape $B^2\le C_f B$, forcing the indices into an ordinary finite interval. They then identify the modeled roots with genuine integers, giving the desired contradiction.

Why the elliptic companion is needed

M2 supplies pointwise rational-point information for individual curves used in existential conditions controlling denominator parity. An average result about almost all twists would not settle those particular curves.

Its strategy arranges finite families of quadratic twists at the vertices of a binary cube. The target curve occupies one vertex. Auxiliary primes make central L-values or derivatives nonzero at the other vertices, with uniformly controlled powers of two. Integral interpolation recovers the missing target value. The construction must preserve bounds as the cube grows and, in rank one, make two related tests succeed simultaneously. This is the role of the arithmetic prime networks described in M2.

Finally, M1 obtains degree four by expressing a polynomial computation as addition and multiplication gates. Each gate gives an equation of degree at most two. Over $\mathbb Q$, a sum of their squares vanishes exactly when every gate equation vanishes, producing the quartic normal form.

7. Scope, limitations, and open questions

Undecidability concerns a single algorithm for all inputs. It does not say that every individual equation is inaccessible, or that useful algorithms for special classes cease to exist. Nor does it establish the same theorem with a fixed number of variables or bounded coefficient sizes.

The argument’s arithmetic dependencies are substantial and are claims in the same collection. They should not be concealed behind the short logical parallel-search argument. The rational-two-torsion companion is narrower than the general BSD statements in family 002, but has a specific role here.

The manuscript leaves existential definability of the integers and many-one completeness outside its conclusions. Determining the sharp restrictions on degree, dimension, and geometric type under which rational solvability becomes decidable remains a separate collection of questions.

8. Sources and further reading

M1’s Sections 1–2 give the finite-test reduction and its arithmetic architecture; Section 4 states and develops the height estimate; Section 7 explains the Turing degree and degree-four conversion. M2’s introduction and Theorem 1.1 state the pointwise 2-converse and describe its interpolation method.

The primary papers linked in Section 3 distinguish the established integer theorem, definability results, and proper-subring results. Poonen’s paper also explains why large subrings and the entire rational field can behave differently. For optional terminology, see Hilbert’s tenth problem and Turing reduction.

Source manuscripts

• M1. OpenAI, Hilbert’s tenth problem over the rational numbers (October 6, 2026).
• M2. OpenAI, A pointwise 2-converse for elliptic curves with rational two-torsion (October 7, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.