On symmetric partial differential operators
Résumé
Let $s 1 ,. .. , s k$ be the elementary symmetric functions of the complex variables $x 1 ,. .. , x k$. We say that $F ∈ C[s 1 ,. .. , s k ]$ is a trace function if their exists $f ∈ C[z]$ such that $F (s 1 ,. .. , s k ] = k j=1 f (x j)$ for all $s ∈ C k$. We give an explicit finite family of second order differential operators in the Weyl algebra $W 2 := C[s 1 ,. .. , s k ] ∂ ∂s 1 ,. .. , ∂ ∂s k$ 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 holonomic system which is a quotient of $W 2$ by an explicit left ideal whose local solutions are linear combinations of the branches of the multivalued root of the universal equation of degree $k: z k + k h=1 (−1) h .s h .z k−h = 0.$
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