Hopkins, MarkKozen, Dexter2007-04-232007-04-231999-01http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.cs/TR99-1724https://hdl.handle.net/1813/7378Parikh'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 $f_i(x_1,\ldots,x_n) \leq x_i$, $1\leq i\leq n$, over a commutative Kleene algebra $K$ has a unique least solution in $K^n$; 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.240918 bytes500240 bytesapplication/pdfapplication/postscripten-UScomputer sciencetechnical reporttechnical report