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

[摘要] In this paper, we show that the truncated binomial polynomials defined by $P_{n, k}(x)=sum^k_{j=0}({nchoose j})x^j$ are irreducible for each $kleq 6$ and every $ngeq k+2$. Under the same assumption $ngeq k+2$, we also show that the polynomial $P_{n, k}$ cannot be expressed as a composition $P_{n, k}(x)=g(h(x))$ with $ginmathbb{Q}[x]$ of degree at least 2 and a quadratic polynomial $hinmathbb{Q}[x]$. Finally, we show that for $kgeq 2$ and $m, ngeq k+1$ the roots of the polynomial $P_{m, k}$ cannot be obtained from the roots of $P_{n, k}$, where $meq n$, by a linear map.

[学科分类] 数学（综合）

[关键词] Truncated binomial expansion;irreducibility;indecomposability.