Seminar of Algebra

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Algebraic and algorithmic study of some generalized functions associated with a real polynomial

Toshinori Oaku (Tokyo Woman's Christian University)
Sala de Seminarios del IMUS
Fri, 12 sep 2014 10:00
Toshinori Oaku is Professor at the Department of Mathematics of Tokyo Woman's Christian University. His research interests lie in the algebraic and algorithmic study of linear PDEs and generalized functions.

For a real polnomial $f$ in several variables, one can define the complex power $f_+^s$ as a distribution (a generalized function) with a holomorphic parameter $s$. Bernstein proved that this distribution can be extended as a distribution-valued meromorphic function of $s$ on the whole complex plane by using what is called the Bernstein-Sato polynomial or the $b$-function, which plays an essential role in $D$-module theory as well as in singularity theory.

Hence it is an interesting problem to determine the poles of $f_+^s$ and the Laurent expansion around each pole. I will present some examples of, as well as algorithms to compute, holonomic systems for Laurent coefficients of $f_+^s$. I will also mention some relation with the local cohomology group associated with $f$.

Oaku 3