Locally finite and locally nilpotent derivations with applications to polynomial flows and polynomial morphisms

Abstract

Abstract.We give a very simple proof of the fact that the Lorenz equations and the Maxwell-Bloch equations do not have a polynomial flow. We also give an algorithm to decide if a two-dimensional vector field over R has a polynomial flow and how to compute the solutions (in case the vector field has a polynomial flow).

Noncommutative symmetric systems over associative algebras

“…t z , which has been studied in [8][9][10]31,30] for the commutative case. The last two components of Ω F t are given directly as…”

A proper 𝐺ₐ action on 𝐶⁵ which is not locally trivial

“…y\ These generators were obtained by implementing a form of the algorithm in [11], easily extended to locally nilpotent, but not necessarily linear, derivations of polynomial rings. It should be noted that van den Essen has given a treatment of the algorithm, suitable for computer implementation, in [3]. Since the latter reference may not be easily accessible, we sketch the application to the example at hand, referring the reader to [11] for details.…”

Triangular derivations related to problems on affine 𝑛-space

“…Locally nilpotent derivations have shown to be very useful in the study of various problems in algebra, algebraic geometry, and differential equations (see [6], [7], [11], [16], [1], [17], [18], [8], [9], and [10]). …”