The projective translation equation and rational plane flows. II. Corrections and additions

Abstract

Abstract. In this second part of the work, we correct the flaw which was left in the proof of the main Theorem in the first part. This affects only a small part of the text in this first part and two consecutive papers. Yet, some additional arguments are needed to claim the validity of the classification results.With these new results in a disposition, algebraic and rational flows can be much more easily and transparently classified. It also turns out that the notion of an algebraic projective flow is a very natural one. For example, we give an inductive (on dimension) method to build algebraic projective flows with rational vector fields, and ask whether these account for all such flows.Further, we expand on results concerning rational flows in dimension 2. Previously we found all such flows symmetric with respect to a linear involution i 0 (x, y) = (y, x). Here we find all rational flows symmetric with respect to a non-linear 1-homogeneous involution i(x, y) = ( y 2 x , y). We also find all solenoidal rational flows. Up to linear conjugation, there appears to be exactly two non-trivial examples.

Planar 2-homogeneous commutative rational vector fields

“…We will need the following result, proved in two independent ways in [5,6]. Birational 1-homogeneous maps (1-BIR for short) R 2 → R 2 were described in [5].…”

“…To find when a vector field (̟, ̺) produces algebraic flow of level 1, we need the following criterion, which follows from results in [6]. Proposition 3.…”

“…On the other hand, a flow φ is algebraic flow of level N if and only if any solution of the same differential equation can be written in the form f (x) = r(x) + σq 1/N (x), where σ ∈ R, r(x) and q(x) are rational functions, and positive N is the smallest possible [6].…”

“…This is a 0-homogeneous function. We know that W satisfies (6). Now, consider a 1-BIR given by ℓ(x, y) = (xA, yA).…”

“…2-homogeneous case. Now, instead of looking at general rational vector fields, in a series of papers [5,6,7,8], and also in [9,10,11,12] (though more general vector fields are dealt with in the last four papers) we embarked on the task to develop the above program in a special case when a vector field is given by a collection of 2-homogeneous rational functions. Indeed, suppose the flow which integrates a fixed n-dimensional rational vector field, is given by the function F (x, z), x ∈ R n , z ∈ R. But now, if the vector field is 2-homogeneous, the time variable can be accommodated within space variables, in a sense that there exists a function φ : R n → R n , such that [5]…”