Let $f\in \mathbb C[x_1, \dots, x_n]$, $n\ge 1$, be a non-constant polynomial. Consider the polynomial $f+t\in \mathbb C[t, x_1,\dots, x_n]$. This is an irreducible polynomial in $\mathbb C(t)[x_1, \dots, x_n]$.

Let $\mathbb C(t)^{alg}$ be an algebraic closure of $\mathbb C(t)$. My question is:

Under which condition $f+t$ remains irreducible in $\mathbb C(t)^{alg}[x_1, \dots, x_n]$ ?

More precisely: we can construct a non-example as follows. If $f=P(g)$ for some $g\in \mathbb C[x_1, \dots, x_n]$ and $P(T)\in \mathbb C[T]$ of degree $\deg P(T)\ge 2$. Then $f+t$ is reducible in $\mathbb C(t)^{alg}[x_1, \dots, x_n]$. My precise question:

Is the above non-example the only one ?

So far I only found a necessary condition for $f+t$ to be reducible over $\mathbb C(t)^{alg}$: if we decompose $f$ into homogeneous components $f=f_d+\cdots +f_1 + f_0$, then there must be a non-constant $h\in \mathbb C[x_1, \dots, x_n]$ such that $h\mid f_{d-1}$ and $h^2\mid f_d$.

**Motivation**: Standard arguments of algebraic geometry show that $f+t$ is irreducible in $\mathbb C(t)^{alg}[x_1, \dots, x_n]$ if and only if for all but finitely many $c\in \mathbb C$, $f+c$ is irreducible in $\mathbb C[x_1,\dots,x_n]$.