Sur les opérateurs différentiels symétriques - Institut Élie Cartan de Lorraine

Date/heure
21 octobre 2019
15:30 - 16:30

Oratrice ou orateur
Daniel Barlet

Catégorie d'évènement
Séminaire de géométrie complexe

Résumé

Let $s_{1}, d o t s, s_{k}$ be the elementary symmetric functions of the complex variables $x_{1}, d o t s, x_{k}$ .
We say that $F i n C [s_{1}, d o t s, s_{k}]$ is a {trace function} if their exists $f i n C [z]$ such that
$F (s_{1}, d o t s, s_{k}] = s u m_{j = 1}^{k} f (x_{j})$ for all $s i n C^{k}$ .
We give an explicit finite family of second order differential operators in the Weyl algebra
$W_{2} := C [s_{1}, d o t s, s_{k}] l a n g l e f r a c p a r t i a l p a r t i a l s_{1}, d o t s, f r a c p a r t i a l p a r t i a l s_{k} r a n g l e$
which generates the left ideal in $W_{2}$ of partial differential operators killing all trace functions.
The proof uses a theorem for symmetric differential operators analogous
to the usual symmetric functions theorem and the corresponding map for symbols. As a corollary, we obtain for each integer $k$
a regular holonomic system which is a quotient of $W_{2}$ by an explicit left ideal whose local solutions are given by linear
combinations of the branches of the multivalued root of the universal equation of degree $k$ :
$z^{k} + s u m_{h = 1}^{k} (- 1)^{h} . s_{h} . z^{k - h} = 0$ .