Parikh's Theorem says that the commutative image of every context free language is the commutative image of some regular set. Pilling has shown that this theorem is essentially a statement about least solutions of polynomial inequalities. We prove the following general theorem of commutative Kleene algebra, of which Parikh's and Pilling's theorems are special cases: Every system of polynomial inequalities $fi(x1,\dots ,xn)\le xi$, $1\le i\le n$, over a commutative Kleene algebra $K$ has a unique least solution in $Kn$; moreover, the components of the solution are given by polynomials in the coefficients of the $fi$. We also give a closed-form solution in terms of the Jacobian matrix.