Irreducibility and Galois groups of truncated binomial polynomials

For positive integers n >= m, let P-n,P-m(x) :=& sum;(m )(j=0)((n) (j)) x(j) = ((n) (0)) + ((n) (1))x + & mldr; + ((n) (m))x(m) be the truncated binomial expansion of (1 + x)(n) consisting of all terms of degree <= m. It is conjectured that for n > m + 1, the polynomial P-n,P-m(x) is irreducible. We confirm this conjecture when 2m <= n < (m + 1)(10). Also we show for any r >= 10 and 2m <= n < (m + 1)(r+1), the polynomial P-n,P-m(x) is irreducible when m >= max{10(6), 2r(3)}. Under the explicit abc-conjecture, for a fixed m, we give an explicit n(0), n(1) depending only on m such that for all n >= n(0), the polynomial P-n,P-m(x) is irreducible. Further for all n >= n(1), the Galois group associated to P-n,P-m(x) is the symmetric group S-m.