# ag.algebraic geometry – Is this property of polynomials generic?

Let $$n geq 2$$, and consider a polynomial $$f$$ in $$n$$ variables, say over a field $$K$$ of characteristic 0. Recall that $$f$$ is geometrically irreducible if $$f$$ is irreducible over the algebraic closure of $$K$$. We know that for $$n geq 2$$, being geometrically irreducible is a generic condition, i.e., applies to a non-empty open subset (in the Zariski topology) of the space of polynomials of degree $$d$$, say.

It seems that some geometrically irreducible polynomials are “more” reducible than others. Here is the example I have in mind: take $$f(x,y) = x^3 – y^2$$, so that $$f$$ is geometrically irreducible. However, $$f(u^2,v^3) = u^6 – v^6$$ IS reducible, in fact splits completely over $$overline{mathbb{Q}}$$.

Let us define two further classes of polynomials: we say that $$f$$ is practically reducible if there exist polynomials $$u_1, cdots, u_n$$ such that $$f(u_1, cdots, u_n)$$ is geometrically reducible, and we say that $$f$$ is algebraically practically reducible if there exist algebraic functions $$u_1, cdots, u_n$$ such that $$f(u_1, cdots, u_n)$$ is a polynomial which is geometrically reducible.

My questions are: is the condition of being “practically irreducible” and “algebraically practically irreducible” generic conditions? That is, do there exist non-empty Zariski-open subsets of polynomials of given degree which are not practically reducible/algebraically practically reducible?

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