Seminario de Álgebra Conmutativa, Geometría Algebraica y Aritmética

Lunes 17 de noviembre de 2025, 11:10h, en el aula 420 del módulo 17, Departamento de Matemáticas UAM

Baldur Sigurðsson (UPM)

Title: On the Jacobian polygon and Łojasiewicz exponent of isolated complex hypersurface singularities

Abstract: A conjecture of Brzostowski, Krasiński and Oleksik predicts a formula for the Łojasiewicz exponent of an isolated Newton nondegenerate hypersurface singularity in terms of its Newton diagram. This result was known in the case of plane curves, and was proved by BKO for surfaces.

We give a counterexample to the conjecture, as it is stated, in four variables. We then present methods which allow us to calculate the Łojasiewicz exponent in terms of a Newton diagram in any dimension. The route we take is to calculate Teissier's Jacobian Newton polygon associated with any Newton nondegenerate function in terms of its Newton diagram. This calculation is simplified by introducing the alternating Jacobian polygon. This allows us to translate the search for the Łojasiewicz exponent to a question on the vanishing of a Newton number.

A crucial technical tool used along the way is a new filtration on a well known Morse-Smale complex which calculates the homology of a Milnor fiber, which we believe to be of independent interest, and applies to any isolated hypersurface singularity.