  For an arbitrary representation $\rho$ of a complex finite-dimensional Lie
algebra, we construct a collection of numbers that we call the Jordan-Kronecker
invariants of $\rho$. Among other interesting properties, these numbers provide
lower bounds for degrees of polynomial invariants of $\rho$. Furthermore, we
prove that these lower bounds are exact if and only if the invariants are
independent outside of a set of large codimension. Finally, we show that under
certain additional assumptions our bounds are exact if and only if the algebra
of invariants is freely generated.
