The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2πic) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture asserts that every such c is a root of the Bernstein-Sato polynomial of the hypersurface. In this note we prove the weak version of the conjecture for hyperplane...

We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety, using the theory of V -filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration of multiplier ideals coincides essentially with the restriction of the V -filtration. This implies a relation between the roots of the Bernstein-Sato polynomial and the jumping coefficients of the multiplier id...

We solve the problem of finding an enclosure for the range of a multivariate polynomial over a rectangular region by expanding the given polynomial into Bernstein polynomials. Then the coefficients of the expansion provide lower and upper bounds for the range and these bounds converge monotonically if the degree of the Bernstein polynomials is elevated. To obtain a faster improvement of the bou...

We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety (which is not necessarily reduced nor irreducible), using the theory of V -filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration by multiplier ideals coincides essentially with the restriction of the V -filtration. This implies a relation between the roots of the Bernstein-Sato polynomia...

Mixture models for density estimation provide a very useful set up for the Bayesian or the maximum likelihood approach. For a density on the unit interval, mixtures of beta densities form a flexible model. The class of Bernstein densities is a much smaller subclass of the beta mixtures defined by Bernstein polynomials, which can approximate any continuous density. A Bernstein polynomial prior i...

The tensorial Bernstein basis for multivariate polynomials in n variables has a number 3n of functions for degree 2. Consequently, computing the representation of a multivariate polynomial in the tensorial Bernstein basis is an exponential time algorithm, which makes tensorial Bernstein-based solvers impractical for systems with more than n= 6 or 7 variables. This article describes a polytope (...