On the distribution of reducible polynomials

Rok, strany: 2009, 349 - 356

Let $\Y_n(t)$ 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 $|\Y_n(t)|$ of the set $\Y_n(t)$ by showing that, as $t\to∞$, $t² log t\ll |\Y₂(t)|\ll t² log t$ and $tⁿ\ll |\Y_n(t)|\ll tⁿ$ for every fixed $n≥ 3$. Further, for $1<{n\over 2}k,n(t)\subset\Y_n(t)$ such that $p(X)\in\Y_k,n(t)$ if and only if $p(X)$ has an irreducible factor in $\Zi[X]$ of degree $k$. Then, as $t\to∞$, we always have $t^k+1\ll |\Y_k,n(t)|\ll t^k+1$ and hence $|\Y_n-1,n(t)|\gg|\Y_n(t)|$ so that $\Y_n-1,n(t)$ is the dominating subclass of $\Y_n(t)$ since we can show that $|\Y_n(t)\setminus\Y_n-1,n(t)|\ll t^n-1( log t)²$. On the contrary, if $R_n^s(t)$ is the total number of all polynomials in $\Y_n(t)$ which split completely into linear factors over $\Zi$, then $t²( log t)^n-1\ll R_n^s(t)\ll t²( log t)^n-1$ $(t\to∞)$ for every fixed $n≥ 2$.

ISO 690:

Kuba, G. 2009. On the distribution of reducible polynomials. In Mathematica Slovaca, vol. 59, no.3, pp. 349-356. 0139-9918. DOI: https://doi.org/10.2478/s12175-009-0131-6

APA:

Kuba, G. (2009). On the distribution of reducible polynomials. Mathematica Slovaca, 59(3), 349-356. 0139-9918. DOI: https://doi.org/10.2478/s12175-009-0131-6