  Let g be a finite-dimensional simple Lie algebra of rank r over an
algebraically closed field of characteristic zero, and let e be a nilpotent
element of g. Denote by g^e the centralizer of e in g and by S(g^e)^{g^e} the
algebra of symmetric invariants of g^e. We say that e is good if the
nullvariety 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 main
result of this paper stipulates that if for some homogeneous generators of
S(g^e)^{g^e}, the initial homogeneous component of their restrictions to e+g^f
are 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 the
above polynomiality condition, in in simple Lie algebras of both classical and
exceptional types. We also give a counter-example in type D_7.
