Differential algebraic dependence and Novikov dependence

TL;DR

This paper introduces new criteria for determining differential algebraic and Novikov dependence of finite systems of elements, providing algorithmic recognition methods in differential polynomial and Novikov algebras.

Contribution

It develops an analogue of Fox derivatives for differential polynomial algebras and establishes criteria for dependence, including algorithmic recognition over characteristic zero fields.

Findings

Established a criterion for differential algebraic dependence.

Proved algorithmic recognizability of dependence in differential polynomial algebras.

Provided a dependence criterion for free Novikov algebras.

Abstract

We define an analogue of the Fox derivatives for differential polynomial algebras and give a criterion for differential algebraic dependence of a finite system of elements. In particular, we prove that differential algebraic dependence of a finite set of elements of a differential polynomial algebra over a constructive differential field $k$ of characteristic zero is algorithmically recognizable. Using a representation of free Novikov algebras by differential polynomials we also give a criterion of Novikov dependence of a finite system of elements of free Novikov algebras.