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Baker–Matiyasevich–Robinson conjecture on Hilbert’s tenth problem for few variables

Establish that, for n ≥ 3 variables, there is no algorithm to decide whether a polynomial equation f(x_1, …, x_n) = 0 with integer coefficients has a positive integer solution; equivalently, prove that the positive-integer-solutions version of Hilbert’s tenth problem remains undecidable in n ≥ 3 variables.

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Background

The paper situates algorithmic questions for curves against the backdrop of undecidability results. While undecidability is known for determining integral solutions in sufficiently many variables (n ≥ 11), the status for fewer variables is conjectural.

The Baker–Matiyasevich–Robinson conjecture posits that undecidability persists for n ≥ 3, at least for the existence of positive integer solutions.

References

Thanks to the negative solution of Hilbert's tenth problem no such algorithm exists for the analogous problem of determining whether there is even a single integral solution to an equation f(x_1, \ldots, x_n) = 0 with f\in Z[x_1, \ldots, x_n] in n\geq 11 variables, and it is a conjecture of Baker, Matiyasevich, and Robinson that the same should hold when n\geq 3, at least with "integral solution" replaced with "positive integral solution".

— Conditional algorithmic Mordell (2408.11653 - Alpöge et al., 21 Aug 2024) in Section “The problem.”