Existence of a universal Diophantine equation of degree 3

Determine whether there exists a universal Diophantine equation of degree 3; that is, construct or refute the existence of a polynomial p(x; n, m) with integer coefficients of total degree 3 such that for every computably enumerable set W there is a tuple of natural numbers m with the property that, for all n ∈ N, n ∈ W if and only if there exists x ∈ N^k satisfying p(x; n, m) = 0.

Background

The paper builds explicit functions with complicated measurability behavior by encoding computably enumerable sets via universal Diophantine equations. Known constructions (e.g., Jones 1982) provide positively universal polynomials of degree 4 (with 58 unknowns) and other trade-offs between degree and number of unknowns.

The author notes that while a degree-3 universal polynomial would not change the degree bound needed for the main measurability independence theorem in this paper, the existence of such a polynomial remains an outstanding problem in the theory of Diophantine representations of computably enumerable sets.

References

It is an open problem whether there is a universal Diophantine equation of degree $3$, but note that having such a polynomial would not decrease the degree needed for \cref{thm:polynomial-Vitali} (since \cref{lem:polynomial-engineering} gives a polynomial of degree $\max{3+\delta,7}$).

— Any function I can actually write down is measurable, right?  (2501.02693 - Hanson, 6 Jan 2025) in Section 5 (Polynomial engineering), preceding Question (quest:degree)