Let $A$ be a ring and $P\in A[X]$ be a monic degree $n$ polynomial.
Let $Disc(P):= Res(P,P')$ be the discriminant of $P$.
If $A=\mathbf Z$, then Minkowski's theorem says that there are no non trivial irreducible polynomials with discriminant equal to $\pm 1$ (the units in $\mathbf Z$).
Since $\mathbf Z$ and $k[Y]$ with $k$ a field have a lot of similarities, one could naively expect that a similar result holds over $k[Y]$. Namely: there are no non-trivial monic (separable) polynomials $P_Y(X) \in (k[Y])[X]$ with discriminant (as a polynomial in $X$) equal to a non zero constant (a unit in $k[Y]$).
This naive statement turns out to be not quite true: If $P(X)\in k[X]$ has non zero discriminant (in $k$) then $P_Y(X,Y):= P(X+Y)$ has same discriminant! (Thus the same holds for $P_Y(X,Y):= P(X+Q(Y))$ for $Q\in k[Y]$ any other polynomial).
However, this seems to be the only bad thing that can happen. For example, for degree 2 or 3 polynomials, one can prove that (in characteristic not $2$ and $3$, say) all polynomials $P_Y(X)$ as above are obtained by translation of a separable polynomial in one variable. For $n=2$ this is trivial; for $n=3$, this amounts to showing that there are no non constant polynomials $A, B \in k[Y]$ such that $A^2+ B^3$ is constant. However, the proofs are really ad hoc and made possible by the simplicity of the expression of the discriminant.
My question thus is :
How to prove that all polynomials $P_Y(X)$ as above are obtained by translation of a polynomial in one variable in degree $>3$. Can we somehow adapt Minkowski's theorem?