Decidability for degree-3 (cubic) Diophantine equations over the integers

Determine whether Hilbert’s Tenth Problem is solvable for the class of Diophantine equations over the integers of total degree δ = 3; that is, decide whether there exists an algorithm that determines, for every integer-coefficient polynomial of degree 3, whether it has an integer solution.

Background

The paper notes established decidability for quadratic (δ ≤ 2) Diophantine equations and for equations in a single variable (ν = 1), highlighting degree 3 as a crucial open threshold.

This problem aims to pin down the exact boundary where decidability transitions from known (quadratic) to unknown (cubic), a central question in the landscape of bounded-complexity versions of Hilbert’s Tenth Problem.