The following question I have asked in MSE, getting one comment. Hopefully, it is ok to ask it here also.

Let $k$ be a field of characteristic zero, $n \in \mathbb{N}$.

**Definitions:**

**(1)** $0 \neq f \in k[x_1,\ldots,x_n]$ is *always irreducible*, if for every $\lambda \in k$, $f+\lambda$ is irreducible in $k[x_1,\ldots,x_n]$.

**(2)** $0 \neq f \in k[x_1,\ldots,x_n]$ is *infinitely irreducible*, if for infinitely many $\lambda \in k$, $f+\lambda$ is irreducible in $k[x_1,\ldots,x_n]$, and call those $\lambda$'s for which $f+\lambda$ is irreducible *good scalars*.

**(3)** $0 \neq f \in k[x_1,\ldots,x_n]$ is *never irreducible*, if there exist no $\lambda \in k$ for which $f+\lambda$ is irreducible in $k[x_1,\ldots,x_n]$.

**Examples:**

**(i)** In $\mathbb{R}[x]$, $x$ is always irreducible, $x^2$ is infinitely irreducible with good scalars $\in (0,\infty)$.

**(ii)** In $\mathbb{C}[x]$, $x$ is always irreducible, $x^2$ is never irreducible.

Question 1:Is it possible to somehow characterize all always irreducible polynomials in $\mathbb{C}[x,y]$?Question 2:Is there a way to distinguish between always irreducibles and infinitely irreducibles?

Examples of always irreducible polynomials in $\mathbb{C}[x,y]$ are:

**(a)** $\lambda x- \mu$, where $\lambda,\mu \in \mathbb{C}$.

**(b)** $\lambda y- \mu$, where $\lambda,\mu \in \mathbb{C}$.

**(c)** $\lambda x + H(y)$, where $\lambda \in \mathbb{C}$, $H(y) \in \mathbb{C}[y]$.

**(d)** $\lambda y + H(x)$, where $\lambda \in \mathbb{C}$, $H(x) \in \mathbb{C}[x]$.

Actually, **(c)** includes **(a)** and **(d)** includes **(b)**.
If I am not wrong, **(c)** and **(d)** can be proved by Eisenstein's criterion. One has to be careful, for example $x+y^2$, in wikipedia's notations we should take $p=x$ not $p=y$.

**(e)** By the fourth answer to this question, $f=g(x)-h(y)$ is irreducible when $\gcd(\deg(g),\deg(h))=1$; in particular, taking $g$ linear yields **(c)**, and taking $h$ linear yields **(d)**.

If I am not wrong, in $k[x,y]$:

If $(f,g)$ is an *automorphic pair*,
then $f$ (and $g$) is always irreducible, where $(f,g)$ is an automorphic pair if $k[x,y]=k[f,g]$ or, equivalently, if $(x,y) \mapsto (f,g)$ is an automorphism of $k[x,y]$.

Moreover, if $(f,g)$ is a *Jacobian pair*,
then $f$ (and $g$) is always irreducible, where $(f,g)$ is a Jacobian pair if $\operatorname{Jac}(f,g):=f_xg_y-f_yg_x$ belongs to $k-\{0\}$.
Indeed, $\frac{k[x,y]}{\langle f \rangle}$ is an integral domain (I can add an argument for this later), so $\langle f \rangle$ is a prime ideal, hence by the second link below, $f$ is irreducible. Repeat this argument for $f + \lambda$ for every $\lambda \in k$, and get that $f + \lambda$ is irreducible for every $\lambda \in k$.

Please see the following related questions: Irreducibility of polynomials in two variables, What do prime ideals in $k[x,y]$ look like?, Irreducibility of Polynomials in $k[x,y]$.

Thank you very much!