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Polynomial Flows in the Plane View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ADVANCES IN MATHEMATICS 55, 173-208 (1985) Polynomial Flows in the Plane HYMAN BASS * Department of Mathematics, Columbia University. New York, New York 10027 AND GARY MEISTERS Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588 Contents. 1. Introduction. I. Polynomial flows are global, of bounded degree. 2. Vector fields and local flows. 3. Change of coordinates; the group GA,(K). 4. Polynomial flows; statement of the main results. 5. Continuous families of polynomials. 6. Locally polynomial flows are global, of bounded degree. II. One parameter subgroups of GA,(K). 7. Introduction. 8. Amalgamated free products. 9. GA,(K) as amalgamated free product. 10. One parameter subgroups of GA,(K). 11. One parameter subgroups of BA?(K). 12. One parameter subgroups of BA,(K). 1. Introduction Let f: Rn + R be a Cl-vector field, and consider the (autonomous) system of differential equations with initial condition x(0) = x0. (lb) The solution, x = cp(t, x,), depends on t and x0. For which f as above does the flow (p depend polynomially on the initial condition x,? This question was discussed in [M2], and in [Ml], Section 6. We present here a definitive solution of this problem for n = 2, over both R and C. (See Theorems (4.1) and (4.3) below.) The main tool is the theorem of Jung [J] and van der Kulk [vdK] * This material is based upon work partially supported by the National Science Foun- dation under Grant NSF MCS 82-02633. 173 OOOl-8708/85 $7.50 Copyright c 1985 by Academic Press. Inc. All rights of reproduction in any form reserved. 174 BASS AND MEISTERS (Theorem (9.1) below) which presents the group GA,(K) of polynomial automorphisms of K* (K = R or C) as an amalgamated free product. There is unfortunately no analogous tool available to treat the case of dimension n > 3, though we can make a few partial observations for n > 3. In particular an even “locally polynomial” flow is global, and globally polynomial of bounded degree (Theorem (4.1) below.) We are grateful to John Morgan for a helpful suggestion in the proof of the latter result. I. POLYNOMIAL FLOWS ARE GLOBAL,• F BOUNDED DEGREE 2. Vector Fields and Local Flows Let K denote either R or C. Let f: W-t K” be a Cl-vector field on some open WC K”. (In case K = C this means that f is C’ interpreted as a function in I?2n.) Consider the autonomous system of differential equations, dx i==f(x) (X=dt . 1 (la) with initial conditions, x(0) = x0 (x0 E WI* (lb) By the usual theorems on the (local) existence and uniqueness of solutions (see, for example [HS], Chapter 8, espeacially Sect. 7) there is a unique solution x = rp(t, x,,) which is defined for t in a maximal open interval J(x,) containing 0 in R. Thus ui(t,x0> =fcPk x0>>, (24 where, as usual, the dot refers to t-derivative, and (2b) Note that f is recovered from cp by the formula, (3) The set Rf= {(t,x)E R x WltEJ(x)} is open in R XK”, and cp:of-+ K” is a Cl-map, called the (local) flow on W defined by the differential Eq. (la), or byf: We also write pi(x) for p(t, x). For each t E R, FLOWS INTHE PLANE 175 p)I is defined on the (possibly empty) open set U, = {x E W 1(t, x) E fif}, which pl maps homeomorphically to the open set V, = cp,(U,) of W. For example it follows from (2b) that, for t = 0, U, = W, and %(X> = x for all x E W. (2b) For s. t E R we have the (local) one parameter group property, (o,o?,(x>)= vls+ l(X) (4) whenever x E U, and q,(x) E U,. In particular (o,(U,) = UP, and v)-~ = p,- ‘. If, for all x0 E W, the maximal interval J(x,) is all of R, then Rf = R x W, and t ++ qt is a homomorphism from iR to the group of homeomorphisms of W, i.e., a one parameter group of homeomorphisms. In this case we call ~1 a global flow. For a discussion of extendability of solutions, and of the distinction between local and global flows, see [BFM 1, [HS], and [GS]. EXAMPLES. The following one dimensional examples show that, even for a polynomial vector field f(x), the flow q(t, x) need not be global (or polynomial): I 176 BASS AND MEISTERS 1. i=f(x)= 1 +x2 x(f) = fp(f, x0) = tan@ + a) a = arctan The horizontal lines represent the f-intervals J(x,) (x0 E I?). 2. a=f(x)=X* -w = & +I> = x0/( 1 - tx,) Here the flow is rational, and can be globalized by passing from the affrne line to the projective line over K. (2.1) QUESTION (cf. [M 11, Section 6). For which C’ vector fields f :K” + K” is the flow rpl(x) a polynomial function of x for each t? (2.2) QUESTION (lot. cit.). Which polynomial transformations of K” can be realized as a qt for some t and f as above? We shall answer these questions for n < 2 by saying what each such f and v, look like in a suitable coordinate system on K”. Before formulating the results, in Section 4, we discuss change of coordinates. FLOWS IN THE PLANE 177 3. Change of Coordinates; the Group GA,,(K) A transformation T: K” + K” is given by coordinate functions, T(x) = (T,(x) ,..., T,(x)), where x = (xl ,..., x,) E K”. It is (K-) differentiable if each Ti is (K-) differentiable. In this case we have the Jacobian matrix D(T)=D(T)(x)= [;l;;; 1;: ;;;;;;j. The Chain Rule takes the form, D(T 0 S) = D(T(S)) = D(T)(S). D(S). (5) When T is invertible and S = T-‘, we find that D(T-‘) =D(T)(T-‘)-I. (6) Let T as above be invertible. Let f be a Cl-vector field on K”, and consider the differential equation i =f(x). (14 The corresponding flow o,(x) is defined by d(y)) =f(cp,(x)). W Put x = T(y) (so that y = T-‘(x)). Then from (la) and the chain rule, f(x) = i = D(T)(y) . y. In the y-coordinates then, (la) takes the form d =g(y> d”W’)W’ +./-V(Y)). (la’) We claim, moreover, that the flow v*(r) defined by (la’) is just the T- conjugate of p, v/t=T-‘orp,oT. (7) Indeed, if vt is defined by (7) then w,(y) = T-‘(rp,(x)), SO d(w,(y)) = D(T-‘)(y,(x)) . d(P$)) dt = WXT-‘h(x)))-’ .f(rp,(x)) (by (la) and (6)) = WXK(Y))-’ .f(T(v,(~))) = g(wt(Y)>* 178 BASS AND MEISTERS (3.1) CONCLUSION. Let T be a K-dtreomorphism of K”. Putting x = T(y), the dtflerential equation, f =f(x), takes the form, i, = g(y), where g(y) = WI(Y)-~ .~(T(Y))), and the flow vt of the latter is T-’ o (pI o T, where v)~ is the jlow of the former. Thus, classifying C’ vector fields f up to change of coordinates in a group G of K-diffeomorphisms of K” is equivalent to classifying (local) flows up to conjugation by elements of G. The transformation T is said to be polynomial if each Ti(x) is a polynomial in x. We can then write T(x) = CrE w a,x’ where, for each r = (r 1,..., r,), xr = xi1 . x:, and a, E K”, with only finitely many a, # 0. Putting lrl=rl+...+rn, the degree of T is then deg(T) = Max{]r] ] a, # O}. We shall be concerned with the group GA,,(K) of polynomial automorphisms of K”, which we call the aflne Cremona group. It consists of polynomial transformations T which have a polynomial inverse. It follows then from the Chain rule (6) that the (polynomial) Jacobian matrix D(T) has a polynomial matrix inverse, so its determinant, being an invertible polynomial, must be a nonzero constant, i.e., an element of KX = K - {O}. From the Chain Rule we conclude: (3.2) LEMMA. The map Ttt Det D(T) is a homomorphism GA,(K)+ KX. Remarks. 1. If T is a polynomial transformation of K” and Det D(T) E KX then, conjecturally, TE GA,(K) (see [BCW] or [M2].) This is the so-called Jacobian conjecture. 2. Every injectiue polynomial transformation T: 6” + C” belongs to GA,(C) (see [BCW].) 3. The group GA,(K) just consists of afline transformations, T(x) = ax + b, a E K ‘, b E K. In contrast, GA,(K) is infinite dimensional for n > 2. 4. Polynomial Flows; Statement of the Main Results Our first theorem, contained in the results ((6.1) and (6.2)) of Section 6, states roughly that if the flow p(t, x) of a smooth vector field f(x) is even locally polynomial in x, then f is polynomial, cp is global, and rp is globally polynomial of bounded degree in x, with coefficients analytic in t. (4.1) BOUNDED GLOBALIZATION THEOREM. Let W be open in K”, and let fl W+ K” be a C’ vector field with jlow cp:Qf + W. Assume that, for some open neighborhood R of {0} x W in Qf, rp(t, x) is polynomial in x on FLOWS INTHEPLANE 179 0, in the following sense: If U, = {x E W 1(t, x) E a} then x M q(t, x) is a polynomial map U, --, K” for each t E R.
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