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 f_i (x₁,...,x_n)⩽x_i, 1⩽i⩽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 f_i. We also give a closed-form solution in terms of the Jacobian matrix of the system