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

In this paper, we show that the truncated binomial polynomials defined by P n , k ( x ) = ∑ j = 0 k n 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 ∈ ℚ [ x ] of degree at least 2 and a quadratic polynomial h ∈ 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.