A Diophantine equation (named after Diophantos of Alexandria) is an equation with one or more integer variables $x,y,z,\ldots$ (often involving also their powers) and with integer coefficients $a,b,c,\ldots$
A Diophantine equation is called linear quadratic, cubic, etc. or, in general, $n$th order) if all variables have at most the power of $1$ ($2$, $3$, etc. $n$).
It is convenient to write a given Diophantine equation in the form $f(x_1,\ldots,x_r)=0$ where $x_1,\ldots,x_r$ are the variables of this equation. The above equation defines interpreted as a function $f:\mathbb Z^r\to \mathbb Z.$ The above examples can be re-written as
Proofs: 1 2 3
Propositions: 4 5 6