Irreducibility and galois groups of truncated binomial polynomials

Abstract

For positive integers $n\ge m$, let $P_{n,m}(x):={\sum }_{j=0}^m\left(\genfrac{}{}{0ex}{}{n}{j}\right)x^j=\left(\genfrac{}{}{0ex}{}{n}{0}\right)+\left(\genfrac{}{}{0ex}{}{n}{1}\right)x+\dots +\left(\genfrac{}{}{0ex}{}{n}{m}\right)x^m$ be the truncated binomial expansion of ${(1+x)}^n$ consisting of all terms of degree $\le m.$ It is conjectured that for $n>m+1$, the polynomial $P_{n,m}(x)$ is irreducible. We confirm this conjecture when $2m\le n<{(m+1)}^{10}.$ Also we show for any $r\ge 10$ and $2m\le n<{(m+1)}^{r+1}$, the polynomial $P_{n,m}(x)$ is irreducible when $m\ge max\{10^6,2r^3\}.$ Under the explicit abc-conjecture, for a fixed $m$, we give an explicit $n_0,n_1$ depending only on $m$ such that $\forall n\ge n_0$, the polynomial $P_{n,m}(x)$ is irreducible. Further $\forall n\ge n_1$, the Galois group associated to $P_{n,m}(x)$ is the symmetric group $S_m.$