Decidability of integer solutions for cubic equations in three or more variables

Determine whether there exists an algorithm that, given any cubic polynomial P(x1, …, xn) with integer coefficients and n ≥ 3, decides whether the Diophantine equation P(x1, …, xn) = 0 has an integer solution; alternatively, prove that no such algorithm can exist for this class of cubic equations.

Background

Hilbert’s tenth problem asks for an algorithm to decide the existence of integer solutions to Diophantine equations; Matiyasevich proved in 1970 that no such general algorithm exists. The undecidability persists for degree-4 equations (for positive solutions), whereas linear and quadratic cases are decidable.

For cubic equations, binary (two-variable) instances are decidable, but for cubic equations in n ≥ 3 variables there is no known decision procedure and no proof of impossibility, leaving the decidability status open.

References

For cubic equations, however, the problem remains open: while cubic equations in at most two variables can be decided algorithmically , for cubic equations in $n\geq 3$ variables there is neither a known algorithm for determining the existence of integer solutions nor a proof that no such algorithm can exist.

— On the shortest open cubic equations  (2603.29831 - Grechuk et al., 31 Mar 2026) in Section 1 (Introduction), preceding and around Problem 1