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

TL;DR

This paper provides simplified proofs of Ritt's zeroes theorem and Seidenberg's embedding theorem, connecting algebraic and analytical approaches to nonlinear PDEs using basic differential algebra and classical PDE existence results.

Contribution

The authors introduce concise proofs of classical theorems in differential algebra, avoiding complex tools and emphasizing elementary methods and the Cauchy-Kovalevskaya theorem.

Findings

New short proofs of Ritt's zeroes theorem and Seidenberg's embedding theorem.

Proofs rely solely on basic differential algebra and classical PDE existence theorems.

Simplification makes these foundational results more accessible.

Abstract

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.