Let $k=\mathbb{A}^1$ be algebraically closed of arbitrary characteristic. I am interested in understanding when a polynomial $f:\mathbb{A}^n\to\mathbb{A}^1$ defines a locally isotrivial family over some cofinite set in $\mathbb{A}^1$.
I know that for $k=\mathbb{C}$ the fibers are generically diffeomorphic. I can easily show that a quasi-homogeneous polynomial defines a trivial family away from $0\in\mathbb{A}^1$. Finally, I also know that for $f:\mathbb{C}^2\to\mathbb{C}$ defined by $f(x,y)=4x^3-y^2$, no two distinct fibers are analytically isomorphic, by the theory of the $j$-invariant.
I have read Serre's 1958 paper where he introduces the notion of isotriviality. I suppose I'll want $H^1(\mathbb{A}^1\setminus\{p_1,\ldots,p_m\},k^\ast)$ for arbitrary $m$ and possibly some groups other than the multiplicative group, but even this eludes me. Any leads?