gpt 6 astra pro attempt on Math-18

Hilbert's Tenth Problem over the Rationals

Definitions and mathematical statement

An input is a finite description of $f\in{\mathbb{Z}}[x_1,\ldots,x_n]$, with no fixed bound on $n$, degree, or coefficient size. Define the decision set \[ \mathcal D_{{\mathbb{Q}}}=\{\langle f\rangle:\exists(q_1,\ldots,q_n)\in{\mathbb{Q}}^n, f(q_1,\ldots,q_n)=0\}. \] Determine whether $\mathcal D_{{\mathbb{Q}}}$ is decidable: one algorithm must halt on every input and return its correct membership bit. Enumeration of rational solutions only semidecides yes-instances and is not a solution. The selected assertion concerns computability, with no polynomial-time demand.

Formulation and concept references: [S1].

Short English statement

Can an algorithm always decide whether a polynomial equation with integer coefficients has a rational solution?

Research attempt

Current frontier.

Research check: 27 September 2026. The 2026 Koymans--Pagano survey states undecidability over infinite finitely generated rings. It does not include ${\mathbb{Q}}$: a ring generated by finitely many rationals inverts only finitely many primes. [S2]. Garcia-Fritz--Pasten--Vidaux prove undecidability when rational equations are augmented with height-comparison predicates, while explicitly leaving the ordinary rational problem open. Those predicates cannot silently be added to this entry. [S3]. The local-global survey describes the positive-existential definability route and its limitations. [S1].

Attack route.

A negative answer could follow from a Diophantine interpretation of integer arithmetic. A positive answer could follow from a total computable bound for the least rational solution. A third idea is to exhaust localizations with finitely many inverted primes. We investigate the exact height-bound equivalence and use explicit equations to stress test the latter two strategies.

Attempt: computable height versus a decision procedure.

Fix a decidable binary encoding of integer polynomials. For $q=a/b$ in lowest terms with $b>0$, set $H(q)=\max(|a|,b)$ and $H(q_1,\ldots,q_n)=\max(1,H(q_1),\ldots,H(q_n))$. The following two statements are equivalent: \[ \mathcal D_{{\mathbb{Q}}}\text{ is decidable}; \tag{21.1} \] \[ \begin{gathered} \text{there is a total computable }B:{\mathbb{N}}\to{\mathbb{N}}\text{ such that}\\ f\in\mathcal D_{{\mathbb{Q}}},\ |\langle f\rangle|\leq L \ \Longrightarrow\ \exists q:\ f(q)=0,\ H(q)\leq B(L). \end{gathered} \tag{21.2} \] If $21.2$ holds, enumerate all reduced fractions with numerator absolute value and positive denominator at most $B(L)$, and all tuples of the required length. Exact rational arithmetic decides whether any is a zero. The list is finite, and $21.2$ justifies returning no at its end.

Conversely, given a decider in $21.1$, list the finitely many valid polynomial encodings of length at most $L$. Decide each. For every positive instance, enumerate rational tuples in increasing height until one solution is found. Each of these finitely many searches terminates; take the maximum of the found heights and 1. This algorithm computes $B(L)$. Neither its running time nor the size of $B(L)$ is controlled. Thus this is a computability equivalence, exactly at the strength of the selected target, not an effective numerical estimate.

Attempt: reduction to quartic sums of squares.

Every polynomial can be evaluated by a finite straight-line computation using addition and multiplication with integer constants. Give each gate an auxiliary variable $z_j$. Its consistency equation has degree at most two: $z_j-z_a-z_b=0$, $z_j-z_az_b=0$, or $z_j-c=0$. Include the equation setting the final value to zero. If the resulting equations are $g_1=\cdots=g_r=0$, replace them by \[ F(x,z)=\sum_{j=1}^r g_j(x,z)^2=0. \tag{21.3} \] Over ${\mathbb{Q}}\subset{\mathbb{R}}$, a sum of squares is zero exactly when every summand is zero. The auxiliary variables are uniquely determined by the inputs in gate order. Hence $F$ has a rational zero if and only if the original polynomial does, and $\deg F\leq4$. This reduction is effective for all finite inputs; a dense or sparse encoding may change its length but not its computability. A decider for this quartic sum-of-squares subclass would therefore decide the original problem. No fixed bound on the number of variables is asserted.

Now test a proposed small height bound on the especially simple family \[ F_r(x_0,\ldots,x_r)=(x_0-2)^2+ \sum_{j=0}^{r-1}(x_{j+1}-x_j^2)^2. \tag{21.4} \] Its unique rational zero has $x_j=2^{2^j}$, by nonnegativity followed by induction. Thus the least multiplicative height is $2^{2^r}$ even though the degree is four and the sparse description has length $O(r\log(r+2))$. In particular no bound of the form $2^{L^A}$ with fixed $A$ works for all these descriptions: $2^r$ eventually exceeds every fixed power of $r\log(r+2)$. This is not evidence that every computable bound fails; a doubly exponential computable bound easily accommodates this family.

Attempt: why finite localizations do not finish the search.

Let $S$ be a finite prime set and $R_S={\mathbb{Z}}[p^{-1}:p\in S]$. If $p\notin S$, the polynomial $px-1$ has the rational zero $1/p$, but no zero in $R_S$, since every reduced denominator in $R_S$ is supported on $S$. Therefore a negative answer for any one finite localization cannot be promoted to a negative answer over ${\mathbb{Q}}$. Exhausting the primes recovers a semidecision for rational solutions, but no stopping rule for negative instances. The fact that ${\mathbb{Q}}$ is a union of these rings does not transfer their undecidability to the union or provide its decider.

For the opposing route, an explicit polynomial $D(t,u_1,\ldots,u_s)$ defining $t\in{\mathbb{Z}}$ by existential rational quantification would let us replace each integer variable of an integer Diophantine equation by its rational counterpart together with $D=0$. Combining all equations as in $21.3$ would yield a reduction of integer undecidability to this target. The needed existential definition has not been constructed here; a definition containing universal quantifiers or height predicates does not meet this requirement. [S1], [S3].

Stress test.

The equivalence $21.1$--$21.2$ bounds at least one point on each nonempty variety, not every point. Finite rational sets and varieties with infinitely many points are handled in exactly the same way. Zero polynomials and constant nonzero polynomials cause no difficulty. No compactness argument has converted pointwise eventual discovery into an effective bound. The quartic reduction uses order only as a proof that rational squares cannot cancel; it does not add an inequality oracle to the input language.

Outcome.

Outcome: Conditional reduction. The attempt gives an exact reduction to quartic sums of squares and an exact computable-minimal-height criterion, with a double-exponential height stress family. These reformulations are not claimed to be new characterizations. Neither the computable bound nor an existential integer interpretation is established. The selected assertion of a total decider remains undecided by this notebook.

Sources

• S1 Sylvy Anscombe, Valentijn Karemaker, Zeynep Kisakürek, Vlerë Mehmeti, Margherita Pagano and Laura Paladino, A survey of local-global methods for Hilbert's Tenth Problem, abstract and Section1. https://arxiv.org/pdf/2309.14987. Catalogue formulation source.
• S2 Hilbert's tenth problem for finitely generated rings. https://arxiv.org/html/2602.04468v1. Catalogue research/status reference; not a proof endorsement.
• S3 Effectivity for existence of rational points is undecidable. https://arxiv.org/abs/2311.01958. Catalogue research/status reference; not a proof endorsement.