Proof mining and effective bounds in differential polynomial rings

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 differentialalgebraic framework of the main results.1 A nonstandard approach. Invent. Math., 76(1): 77-91, 1984.

Explicit polynomial bounds on prime ideals in polynomial rings over fields

“…Remark 3.4. In [27], Theorem 2.5, we call this fact "internal flatness" because it is equivalent to flatness of extensions of polynomial rings by internal polynomial rings in the sense of nonstandard analysis. [2] refers to such results as "effective flatness".…”

“…Abstract results from proof theory [17,29] suggest that it is possible to extract explicit bounds from proofs that use ultraproducts. The authors demonstrated [27] that such methods can be used to extract explicit bounds from ultraproduct proofs in commutative and differential algebra [14] [30]. (For instance, in Theorem 2.8 and Lemma 7.7 of [27] we gave another bound on b(n, d), but that bound was not polynomial in the degree of the generators.)…”

“…The authors demonstrated [27] that such methods can be used to extract explicit bounds from ultraproduct proofs in commutative and differential algebra [14] [30]. (For instance, in Theorem 2.8 and Lemma 7.7 of [27] we gave another bound on b(n, d), but that bound was not polynomial in the degree of the generators.) However the arguments here, while inspired by Schmidt-Göttsch's approach, were not purely extracted by proof theory; instead we bring in known results using Gröbner bases, etc., where possible and add new arguments to simplify (or clarify) the most complicated parts of Schmidt-Göttsch's argument.…”

Rates of convergence for asymptotically weakly contractive mappings in normed spaces

“…A comprehensive background to the area is presented in [17], and the survey papers [18,24] provide a good overview of some of the many applications in analysis. Proof mining continues to be expanded to new areas of mathematics: In recent years this includes pursuit-evasion games [23], differential algebra [29] and Tauberian theory [28].…”

Bounds for Indexes of Nilpotency in Commutative Ring Theory: A Proof Mining Approach

“…The program was initiated in earnest by Ulrich Kohlenbach (who laid out its theoretical basis in [12]) and has been quite successful in some parts of analysis (see the book [13] and the more recent [15]). Recently, it was also used successfully in algebra [18] to extract uniform bounds whose existence was first established using nonstandard methods in [20] and [10].…”