  Let s 1 ,. .. , s k be the elementary symmetric functions of the complex
variables x 1 ,. .. , x k. We say that F $\in$ C[s 1 ,. .. , s k ] is a trace
function if their exists f $\in$ C[z] such that F (s 1 ,. .. , s k ] = k j=1 f
(x j) for all s $\in$ C k. We give an explicit finite family of second order
differential operators in the Weyl algebra W 2 := C[s 1 ,. .. , s k ]
$\partial$ $\partial$s 1 ,. .. , $\partial$ $\partial$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.
