An estimate of the roots of b-functions by Newton polyhedra
By \textit{I. N. Bernshtejn} [Funkts. Anal. Prilozh. 6, Nr. 4, 26-40 (1972; Zbl 0282.46038)] and \textit{J.-E. Björk} [Rings of differential operators (1979; Zbl 0499.13009)], any \(f\in {\mathcal O}\), the set of germs of holomorphic functions at the origin of \(C^{n+1}\), there exist P(s)\(\in {\mathcal D}[s]\), \({\mathcal D} ={\mathcal O}[\partial /\partial x_ 0,...,\partial /\partial_ n]\) such that \(P(s)f^{s+1}=b(s)f^ s\). Making use of the theory of b-functions by \textit{T. Yano} [Publ. Res. Inst. Math. Sci. 14, 111-202 (1978; Zbl 0389.32005)], the author proves that any root of \(b(s)/(s+1)\geq 1/\inf \{t;\quad(t,t,...,t)\in \Gamma_+(f)\}\) when f has an isolated singularity and is nondegenerated.
scientific article; zbMATH DE number 3849671