Rules for whether an $n$ degree polynomial is an $n$ degree power Given an $n$ degree equation in 2 variables ($n$ is a natural number)
$$a_0x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_{n-1}x+a_n=y^n$$
If all values of $a$ are given rational numbers, are there any known minimum or sufficient conditions for $x$ and $y$ to have:


*

*Real number

*Rational number

*Integer


solutions and how many of them would exist. If it is not known/possible (or too hard) for an $n$ degree polynomial, do such conditions exist for quadratic ($n=2$) and cubic ($n=3$) polynomials.

This addresses user45195's question and is too long for a comment. 
When I said too broad, it was because the question originally didn't limit the field of $x$. A familiar field is the complex numbers $\mathbb{C}=a+bi$, of which a special case are the reals $\mathbb{R}$, and even more limited, the rationals $\mathbb{Q}$. 
If $x$ was in $\mathbb{C}$, then it's just an old result (Fundamental Theorem of Algebra), that for any $y$, one can always find $n$ roots $x$ that solve the equation in your post, and there's nothing new to be said.
However, if $x,y$ are limited to the rationals $\mathbb{Q}$, that's where it gets interesting. The equation,
$$f(x) = y^2\tag1$$
where the degree $d$ of $f(x)$ is either $d = 2,3,4,5$ has been extensively studied. See algebraic curve, including Pell equations ($d=2$), elliptic curve ($d=3,4$), and hyperelliptic curve ($d=5$). For,
$$f(x) = y^3\tag2$$
then $d=3$ still has special cases as elliptic curves. For $d=4$, see trigonal curve. However, for the higher, 
$$f(x) = y^m\tag3$$
where $d,m>3$, is more complicated. See superelliptic curve.
