Proofmap
Family 004Number theory62 pp · Oct 6, 2026

Hilbert's tenth problem over the rational numbers

What this paper proves

There is no algorithm that, given a polynomial f∈Z[X1,…,Xn]f\in\mathbb{Z}[X_1,\ldots,X_n] with nn part of the input, decides whether ff has a zero in Qn\mathbb{Q}^n. The proof reduces integer solvability to rational solvability through finite systems of rational constraints built from ff, and shows that if all of them are solvable then ff has an integer zero, using a formula that rules out odd poles and a height bound from odd contact with five points. Since integer solvability is undecidable, this answers the rational analogue of Hilbert's tenth problem negatively.

Main theorem

theoremnovelHilbert's tenth problem over Q is undecidablethm:main
No algorithm can take a polynomial f∈Z[X1,…,Xn]f\in\Z[X_1,\ldots,X_n], with nn part of the input, and decide whether ff has a zero in Qn\Q^n.
Theorem 1.1 · as stated in the paper
There is no algorithm which, given a polynomial f∈Z[X1,…,Xn]f\in\Z[X_1,\ldots,X_n], with nn part of the input, decides whether ff has a zero in Qn\Q^n.

Where the new idea lives · 5 novel steps (one is the main theorem)

Theorem 2.4Height bounded by radical of contact primes
Bounds height of ss by HMS(s)cH M_S(s)^c off five exceptional points
Proposition 2.8Positive existential formula forcing even poles
Gives positive existential Φ(b)\Phi(b) true on integers, forcing even poles
Proposition 2.12Finite test success gives integer zero
If all finite rational tests for ff are solvable, ff has a zero in Zn\mathbb Z^n
Lemma 6.1Prime patterns with controlled reduction orders
Finds thirteen numbers ϵkr\epsilon kr with large good primes rr, bounded kk

Shape of the proof

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