Dimension de Krull, Nullstellens\"atze et \'Evaluation Dynamique

TL;DR

This paper provides a constructive Nullstellensatz linking algebraic identities with the impossibility of certain increasing sequences of varieties, offering a new elementary characterization of Krull dimension using dynamical algebraic structures.

Contribution

It introduces a constructive Nullstellensatz and a novel elementary characterization of Krull dimension via pseudo regular sequences using dynamical algebraic structures.

Findings

Constructive Nullstellensatz relating algebraic identities and variety sequences

New characterization of Krull dimension through pseudo regular sequences

Application of dynamical algebraic structures in algebraic geometry

Abstract

We prove constructively a Nullstellensatz giving an equivalence between the existence of a certain kind of algebraic identity on one hand, and the impossibility of finding an increasing sequence of irreducible varieties obeying certain constraints on the other hand. The ususal Nullstellensatz corresponds to the case of varieties that are reduced to a point. We settle also a similar formal Nullstellensatz related to increasing sequences of primes. An important particular case is given by the notion of pseudo regular sequence. This allows a new characterisation of the Krull dimension of a ring. This characterisation via pseudo regular sequences is elementary and constructive. Our method uses dynamical algebraic structures which were introduced in previous papers.