Parikh's Theorem in Commutative Kleene Algebra

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 finite system of polynomial inequalities fi x xn xi, i n, over a commutative Kleene algebraK has a unique least solution in K; 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 of the system.