Computationally classifying polynomials with small Euclidean norm having reducible non-reciprocal parts
Volume 118 / 2019
Abstract
Let $f(x)$ be a polynomial with integer coefficients. If either $f(x) = x^{{\rm deg}\,{f}}f(1/x)$ or $f(x) = -x^{{\rm deg}\,{f}}f(1/x)$, then $f(x)$ is called reciprocal. We refer to the non-reciprocal part of $f(x)$ as the polynomial $f(x)$ removed of each of its irreducible reciprocal factors in ${\mathbb Z}[x]$ with a positive leading coefficient. In $1970$, Schinzel proved that for a given collection of $r + 1$ integers $a_0,\dots ,a_r$ it is possible to classify the positive integers $d_1,\dots ,d_r$ for which the non-reciprocal part of $a_0 + a_1x^{d_1} + .\kern1.3pt.\kern1.3pt. + a_rx^{d_r}$ is reducible. Specific classification results have been given by Selmer, Tverberg, Ljunggren, Mills, Schinzel, Solan, and the first author. We extend an approach of the first author to complete a similar classification for all polynomials with norm $\le {5}^{1/2}$ and some additional sparse polynomials.
