# On the distribution of reducible polynomials

In: Mathematica Slovaca, vol. 59, no. 3
Gerald Kuba

## Details:

Year, pages: 2009, 349 - 356
Keywords:
lattice points, true order of magnitude of counting functions
Let $\Yn(t)$ denote the set of all reducible polynomials $p(X)$ over $\Zi$ with degree $n≥ 2$ and height $≤ t$. We determine the \textit{true order of magnitude} of the cardinality $|\Yn(t)|$ of the set $\Yn(t)$ by showing that, as $t\to∞$, $t2 log t\ll |\Y2(t)|\ll t2 log t$ and $tn\ll |\Yn(t)|\ll tn$ for every fixed $n≥ 3$. Further, for $1<{n\over 2}k,n(t)\subset\Yn(t)$ such that $p(X)\in\Yk,n(t)$ if and only if $p(X)$ has an irreducible factor in $\Zi[X]$ of degree $k$. Then, as $t\to∞$, we always have $tk+1\ll |\Yk,n(t)|\ll tk+1$ and hence $|\Yn-1,n(t)|\gg|\Yn(t)|$ so that $\Yn-1,n(t)$ is the dominating subclass of $\Yn(t)$ since we can show that $|\Yn(t)\setminus\Yn-1,n(t)|\ll tn-1( log t)2$. On the contrary, if $Rns(t)$ is the total number of all polynomials in $\Yn(t)$ which split completely into linear factors over $\Zi$, then $t2( log t)n-1\ll Rns(t)\ll t2( log t)n-1$ $(t\to∞)$ for every fixed $n≥ 2$.