Bounds for Hilbert coefficients (Q5918493)

Let \((R,\mathfrak{m})\) be a noetherian local ring of Krull dimension \(d\), and \(\mathfrak{a}\) an \(\mathfrak{m}\)-primary ideal of \(R\), i.e. \(\sqrt{\mathfrak{a}}=\mathfrak{m}\). The Hilbert-Samuel function \(H_{\mathfrak{a}}(-):\mathbb{Z}\rightarrow \mathbb{N}_{\geq 0}\) of \(R\) with respect to \(\mathfrak{a}\) is given by \(H_{\mathfrak{a}}(n)=\ell_{R}(R/\mathfrak{a}^{n})\) for every \(n\geq 0\) and \(H_{\mathfrak{a}}(n)=0\) for every \(n< 0\). There exists a unique polynomial \(P_{\mathfrak{a}}(X)\in \mathbb{Q}[X]\) of degree \(d\) such that \(H_{\mathfrak{a}}(n)=P_{\mathfrak{a}}(n)\) for \(n\gg 0\) which is written as \[P_{\mathfrak{a}}(n)=\sum_{i=0}^{d}(-1)^{i}\binom{n+d-i-1}{d-i}e_{i}(\mathfrak{a})\] where \(e_{i}(\mathfrak{a})\)'s are called the Hilbert coefficients of \(\mathfrak{a}\). On the other hand, Elias has shown that \(\mathrm{depth}_{R}\left(\mathfrak{m},\mathrm{gr}(\mathfrak{a}^{k})\right)\) is constant for \(k\gg 0\), and denoted this constant by \(\sigma(\mathfrak{a})\). In the paper under review, the authors prove the non-positivity of the Hilbert coefficients \(e_{i}(\mathfrak{a})\) under some conditions for \(\sigma(\mathfrak{a})\). In the special case where \(\mathfrak{a}\) is a parameter ideal, i.e. an ideal generated by a system of parameters for \(R\), they establish some bounds for the Hilbert coefficients of \(\mathfrak{a}\) in terms of the Krull dimension \(d\) and the first Hilbert coefficient \(e_{1}(\mathfrak{a})\).