Proof mining and effective bounds in differential polynomial rings

TL;DR

This paper applies proof-theoretic methods to extract effective bounds from nonconstructive proofs in polynomial and differential polynomial rings, providing new insights and bounds in differential algebra.

Contribution

It introduces a novel proof mining approach to obtain effective bounds in differential polynomial rings, including new bounds not previously documented.

Findings

New effective bounds in differential polynomial rings

Analysis of the constructive content of Noetherian rings

Effective bounds related to the Nullstellensatz in differential settings

Abstract

Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature, from the resulting proofs. In the process we discuss the constructive content of Noetherian rings and the Nullstellensatz in both the classical and differential settings. Sufficient background is given to understand the proof-theoretic and differential-algebraic framework of the main results.