Some irreducibility and indecomposability results for truncated binomial polynomials of small degree

Abstract

In this paper, we show that the truncated binomial polynomials defined by $P_{n,k}(x)={\sum }_{j=0}^{k} {n \choose j} x^{j}$ are irreducible for each k≤6 and every n≥k+2. Under the same assumption n≥k+2, we also show that the polynomial P _n,k cannot be expressed as a composition P _n,k (x) = g(h(x)) with $g \in \mathbb {Q}[x]$ of degree at least 2 and a quadratic polynomial $h \in \mathbb {Q}[x]$. Finally, we show that for k≥2 and m,n≥k+1 the roots of the polynomial P _m,k cannot be obtained from the roots of P _n,k , where m≠n, by a linear map.