Diophantine set in the context of Coefficient

In mathematics, a Diophantine equation is an equation of the form P(x₁, ..., x_j, y₁, ..., y_k) = 0 (usually abbreviated P(x, y) = 0) where P(x, y) is a polynomial with integer coefficients, where x₁, ..., x_j indicate parameters and y₁, ..., y_k indicate unknowns.

A Diophantine set is a subset S of ${\displaystyle \mathbb {N} ^{j}}$, the set of all j-tuples of natural numbers, so that for some Diophantine equation P(x, y) = 0,