For positive integers $n>k$, let $P_{n,k}(x)=\displaystyle\sum_{j=0}^k
\binom{n}{j}x^j $ be the polynomial obtained by truncating the binomial
expansion of $(1+x)^n$ at the $k^{th}$ stage. These polynomials arose in the
investigation of Schubert calculus in Grassmannians. In this paper, the authors
prove the irreducibility of $P_{n,k}(x)$ over the field of rational numbers
when $2\leqslant 2k \leqslant n<(k+1)^3$
Let g x be a monic irreducible defectless polynomial over a henselian valued field K v , i.e., K is a defectless extension of K v for any root of g x . It is known that a complete distinguished chain for with respect to K v gives rise to several invariants associated with g x . Recently Ron Brown studied certain invariants of defectless polynomials by introducing strict systems of polynomial extensions. In this article, the authors establish a one-to-one correspondence between strict systems of polynomial extensions and conjugacy classes of complete distinguished chains. This correspondence leads to a simple interpretation of various results proved for strict systems. The authors give new characterizations of an invariant g introduced by Brown.
Let (K, v) be a henselian valued field of arbitrary rank which is not separably closed. Let k be a subfield of K of finite codimension and v k be the valuation obtained by restricting v to k. In this paper, we give some necessary and sufficient conditions for (k, v k ) to be henselian. In particular, it is shown that if k is dense in its henselization, then (k, v k ) is henselian. We deduce some well known results proved in this direction through other considerations.
It is well known that if f (x) is a monic irreducible polynomial of degree d with coefficients in a complete valued field (K, | |), then any monic polynomial of degree d over K which is sufficiently close to f (x) with respect to | | is also irreducible over K. In 2004, Zaharescu proved a similar result applicable to separable, irreducible polynomials over valued fields which are not necessarily complete. In this paper, the authors extend Zaharescu's result to all irreducible polynomials without assuming separability.
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