From algebra to analysis: new proofs of theorems by Ritt and Seidenberg

Ritt's theorem of zeroes and Seidenberg's embedding theorem are classical results in differential algebra allowing to connect algebraic and model-theoretic results on nonlinear PDEs to the realm of analysis. However, the existing proofs of these results use sophisticated tools from constructive algebra (characteristic set theory) and analysis (Riquier's existence theorem). In this paper, we give new short proofs for both theorems relying only on basic facts from differential algebra and the classical Cauchy-Kovalevskaya theorem for PDEs.Comment: 13 page

On the Coalgebra of Partial Differential Equations

We note that the coalgebra of formal power series in commutative variables is final in a certain subclass of coalgebras. Moreover, a system Sigma of polynomial PDEs, under a coherence condition, naturally induces such a coalgebra over differential polynomial expressions. As a result, we obtain a clean coinductive proof of existence and uniqueness of solutions of initial value problems for PDEs. Based on this characterization, we give complete algorithms for checking equivalence of differential polynomial expressions, given Sigma

{An orderly linear PDE system with analytic initial conditions with a non analytic solution}

Special Issue on Computer Algebra and Computer AnalysisInternational audienceWe give a linear PDE system, with analytic initial conditions given w.r.t an orderly ranking, the solution of which is not analytic (moreover the solution is not Gevrey for any order). This examples proves that the analyticity Riquier theorem (generalization of the Cauchy-Kovalevskaya theorem) does not generalize to PDE systems endowed with orderly rankings