Decidability for Diophantine equations with a small number of integer unknowns (2–10)

Ascertain the decidability status of Hilbert’s Tenth Problem for Diophantine equations over the integers with a bounded number of unknowns ν in the range 2 ≤ ν ≤ 10; specifically, determine for each ν whether there exists an algorithm that decides solvability of all integer-coefficient polynomial equations with at most ν unknowns.

Background

Beyond the single-variable case (ν = 1), the decidability of Hilbert’s Tenth Problem for higher but small numbers of unknowns remains largely unresolved, as explicitly stated by the authors.

Resolving these cases would clarify the complexity landscape for bounded-unknown formulations of the problem and complement advances in universality and undecidability at larger bounds.