Remarks on Eisenstein (Q2861572)

Let \(K\) be a number field of degree \(d\) and \(M_K\) the set of absolute values of \(K\) normalized to extend the standard absolute values of \(Q\). A \(M_K\)-divisor \({\mathcal A} = (A_v)_{v\in M_K}\) is a real-valued function of \(M_K\) which in every \(v\in M_K\) associates a positive number \(A_v\) such that for all but finitely many \(v\) we have \(A_v = 1\). The height of the \(M_K\)-divisor \(\mathcal A\) is defined to be the quantity NEWLINE\[NEWLINEh({\mathcal A}) = \frac{1}{d} \sum_{v\in M_K} d_v \max\{\log A_v,0\},NEWLINE\]NEWLINE where \(d_v\) is the absolute local degree of \(v\). Let \(P(z,w)\in K[z,w]\) be an irreducible polynomial and \(f(z) = \sum_{k=0}^{\infty} a_kz^k\) a power series with coefficients in the algebraic closure \(\bar{K}\) of \(K\) satisfying \(P(z,f(z))=0\). Then a classical theorem of Eisenstein asserts that there is a \(M_K\)-divisor such that \(|a_k|_v \leq A_v^{k+1}\) \((k=0,1,\ldots)\) for every \(v \in M_k\).NEWLINENEWLINEIn this paper, an explicit quantitative version of the above result is given improving previous results. More precisely, the main result of this paper is the following: If \(P(z,w)\) and \(f(z)\) are as above, then there exist effective \(M_K\)-divisors \({\mathcal A}^{\prime} = (A^{\prime})_{v\in M_K}\) and \({\mathcal A} = A_{v\in M_K}\) such that NEWLINE\[NEWLINE|a_k|_v \leq A_v^{k+1} \;\;\;(k=0,1,\ldots)NEWLINE\]NEWLINE for any \(v\in M_K\) anyhow extended to \(\bar{K}\) and such that NEWLINE\[NEWLINEh({\mathcal A}^{\prime}) \leq h_p(P)+\log 3, \quad h({\mathcal A}) \leq (3n-1)H_p(P)+3n\log(mn)+7n,NEWLINE\]NEWLINE where \(m= \deg_zP\), \(n = \deg_wP\) and \(h_p(P)\) the projective height of \(P(z,w)\). Moreover, the authors obtain more general results on ramified series, Laurent series and compute an explicit upper bound for the discriminant of the number field generated by the coefficients of the series \(f(z)\).