Density of multivariate integer polynomials that are irreducible This is possibly well-known, but I couldn't find any references for it.
Let $d,n \geq 2$ and let $R$ be the set of homogeneous polynomials of degree $d$ in $n$ variables with coefficients in $\mathbb{Z}$. For $f \in R$, we define the height $H(f)$ of $f$ as the maximum of the absolute values of the coefficients of $f$.

Does $\lim\limits_{B \to \infty} \frac{|\{f \in R | f \text{ is irreducible, } H(f) \leq B\}|}{|\{f \in R |  H(f) \leq B\}|}$ exist? If so, what is it?

Many thanks in advance!
 A: If by irreducible you mean irreducible over $\mathbf{Q}$, then the limit will certainly be $1$, that is, almost all such polynomials will be irreducible.
Let $f(x_1,\ldots,x_n)$ be a degree $d$ homogeneous polynomial. If the specialization $h(x,y) = f(x,y,0,0,\ldots,0)$ is irreducible of degree $d$, then $f$ will also be irreducible. This reduces the problem to the case of polynomials in one variable, since $h(x,y) = y^d g(x/y)$ where $g(t)$ is a polynomial of degree $d$.
The main point is now that (for your search box) when $B$ gets large, the reduction of $g(t)$ modulo a fixed prime $p$ is more or less a random polynomial of degree $\le d$ modulo $p$. The probability that such a polynomial will be irreducible (and degree $d$) is easy enough to compute exactly, and is asymptotically $(1-1/p)/d$ as $p$ increases. But is is certainly at least $1/(2d)$ for $d$ large. Moreover, when $B \gg e^X$ increases the distributions modulo all primes $\ll X$ will be independent, and so the probability that it will be reducible for all primes in this range is at most
$$\left(1 - \frac{1}{2d}\right)^{\pi(X)}$$
which clearly tends to $0$ as $X$ tends to infinity and the number of primes increases without bound.

A more sophisticated (at least superficially) answer is to use effective versions of Hilbert irreducibility. Since the generic degree $d$ polynomial is irreducible almost all specializations will be so as well; you can probably show at least that the number of exceptions is at most $1/\sqrt{B}$ of the total number of polynomials. I guess this is probably optimal for $d=n=2$.
