Bernstein polynomials of a smooth function restricted to an isolated hypersurface singularity
Set \(\mathcal{O}=\mathbb{C} \{x_1,x_2,\ldots,x_n\}\) the ring of germs at \(0\) of complex holomorphic functions, \(g\in \mathcal{O}\), such that \(g(0)=0\), and \(\mathcal{R}= \mathcal{O}[1/g] / \mathcal{O}\) the local cohomology module with support in the hypersurface \((X,0)\) defined by \(g\). For a germ of function \(f \in \mathcal{O}\) nonzero on \(X\), there are functional equations in \( \mathcal{R}\otimes \mathcal{O}[1/f,s] f^s\) of the form \(b(s)\delta f^s=P \cdot \delta f^{s+1}\). The unitary generator of the ideal of polynomials \(b(s)\) is called to be the Bernstein polynomial of \(f\) associated with \(\delta\). It is well known that roots of these polynomials determine the eigenvalues of the monodromy of the restriction \(f| _X :(X,0) \rightarrow (\mathbb{C},0)\). The author studies these polynomials in the particular case where \(f\) is a germ of smooth function and gives an algorithm to compute them when \(f\) is a coordinate and \(g\) in non-degenerate with respect to its Newton boundary.
scientific article; zbMATH DE number 2113015
scientific article