Polynomial flows in the plane

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

“…The vector field V is then called a poly-flow vector field, abbreviated p-f vector field. It is shown by Bass and Meisters in [2] that if F is a p-f vector field, then the C1 vector field V itself is polynomial, the solutions are defined for all t £ R, and tp' has a bounded degree in x, independent of t. Using these results, Coomes and Zurkowski give a purely algebraic description of a p-f vector field on Rn (or C").…”

“…Observe that RD is a /c-sublagebra of R. To describe the second principle we assume that R = ©"6Zi?« is a graded /^-module. Hence each element g £ R can be written uniquely as a sum of First we recall some facts on polynomial flows (see [9,2,5]). Let V be a C1 vector field on R" , and consider the initial value problem (2.1) y = V{y), y(0) = XGR".…”

Polynomial flows on 𝐶ⁿ

“…\r\<d (See [BM,Theorem 6.1].) Denote the right-hand side of (1.2) by 4> (t, x Since 4> is polynomial in x for each t, it extends uniquely to a function analytic on W x C" .…”

On the torsion part of 𝐶^{[𝑛]} with respect to the action of a derivation

“…Coomes and Zurkowski [8] show that 0 is a polynomial flow if and only if T(D) = C[n]. (See [3,5,6,7,10,11,12,13,14,15,17] for other results about polynomial flows.) Since the torsion part T(D) can play an important role in determining whether a vector field has a polynomial flow, we wish to gain a deeper understanding of the properties of T(D).…”