On Brown polynomials. II (Q2282616)

Let \(v\) be a valuation of a field \(K\) with value group \(G\) and valuation ring \(R\) having maximal ideal \(\mathcal M\). Let \(v^x\) denote the Gaussian prolongation of \(v\) to the rational function field \(K(x)\) defined by \(v^x(\sum_{i}a_ix^i)= \min_i\{v_p(a_i)\}, a_i\in K\). Let \(\phi(x)\in R[x]\) be a monic polynomial which is irreducible module \(\mathcal M\). Given a positive element \(\lambda\) of the divisible closure of \(G\) and some integer \(e>0\), a polynomial \(f(x)\in R[x]\) called a Brown polynomial of the type \((\phi,~e,~\lambda)\) if the \(\phi\)-expansion of \(f\) given by \(f=\sum_{i}f_i\phi^i\), \(f_i\in R[x]\) with deg \(f_i<\) deg \(\phi\) satisfies the following two conditions: \begin{itemize} \item[(i)] \(v^x(f_e)=0,~v^x(f_0)=e\lambda\), \item[(ii)] \(v^x(f_i)\geq (e-i)\lambda\) for \(0< i < e\). \end{itemize} In this paper, it is proved that the product of two Brown polynomials and a factor of a Brown polynomial is again a Brown polynomial. It is also shown that when \((K,~v)\) is Henselian, then any monic irreducible polynomial belonging to \(R[x]\) is a Brown polynomial. The proofs are well explained; the paper extends some already known results in this direction. For Part I see Sib. Math. J. 53, No. 4, 656--658 (2012); translation from Sib. Mat. Zh. 53, No. 4, 819--821 (2012; Zbl 1275.12002).