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Let g be a finite-dimensional simple Lie algebra of rank r over analgebraically closed field of characteristic zero, and let e be a nilpotentelement of g. Denote by g^e the centralizer of e in g and by S(g^e)^{g^e} thealgebra of symmetric invariants of g^e. We say that e is good if thenullvariety of some r homogeneous elements of S(g^e)^{g^e} in the dual of g^{e}has codimension r. If e is good then S(g^e)^{g^e} is polynomial. The mainresult of this paper stipulates that if for some homogeneous generators ofS(g^e)^{g^e}, the initial homogeneous component of their restrictions to e+g^fare algebraically independent, with (e,h,f) an sl2-triple of g, then e is good.As applications, we obtain new examples of nilpotent elements that verify theabove polynomiality condition, in in simple Lie algebras of both classical andexceptional types. We also give a counter-example in type D_7.

The symmetric invariants of the centralizers and Slodowy grading