Is an equation containing a factorial still called "Diophantine equation"? 
Is an equation like $x^3+y^4=z!$ , $x,y,z\in\mathbb Z$ , $z\ge 0$ called "Diophantine equation" ?

I am not sure whether the only requirement is that the variables must be integers. Are factorials allowed, and what about something like $x^3+y^4=2^z$ ?
An additional question :
Is it right that the negative result of Hilbert's tenth problem (there is no general method to decide whether a diophantine equation is solveable in the integers) is already valid if we only allow polynomial equations ?
 A: As you can see, different people have different opinions about what a "Diophantine equation" is. E.g., for a long time, some people meant rational solutions to equations, and others meant integer. These are significantly different, since, unsurprisingly, classification of integer solutions is much more complicated than classification of rational solutions. 
And, indeed, whether asking for integer solutions or rational, the variously more-or-less restrictive types have different properties, as we know by this year. For example, I think all of the modern-times results that use (modern) algebraic geometry can only bear on polynomial equations, for obvious reasons, whether addressing rational or integer solutions. (Mordell-Weil, Faltings, et al).
But in a broad sense, especially if asking for integer solutions, a broad sense of "Diophantine" can certainly allow "reasonable" operations on integers. The question of whether there is any systematic machinery that might conceivably bear on such questions depends enormously on what, if any, larger context such a question could be imbedded.
