On symmetric partial differential operators

Let sigma(1),..., sigma(k) be the elementary symmetric functions of the complex variables x(1),..., x(k). We say that F epsilon C[ sigma 1,..., sigma(k)] is a trace function if their exists f epsilon C[z] such that F(sigma(1) ,..., sigma(k)) = Sigma(k)(j=1) f (x (j)) for all sigma epsilon C-k. We give an explicit finite family of second order differential operators in the Weyl algebra W2 := C[sigma(1),..., sigma(k)] partial derivative/partial derivative sigma(1) ,....,partial derivative/partial derivative sigma(K) which generates the left ideal in W2 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 an application, 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) + Sigma(k)(h=1)(-1)(h) sigma(h)Z(k-h) = 0.