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Polynomial flows, symmetry groups and conditions sufficient for injectivity of maps
Abstract
Consider the initial value problem(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\dot y (\equiv {dy\over dt}) = {\bf V}(y), y(0) = x \in {\bf R}\sp{n}\eqno(1)$$(TABLE/EQUATION ENDS)where V is a $C\sp1$ vector field on R$\sp{n}$ and for each t the solution (flow) $\phi$ is a polynomial function of the initial condition x. Call such a vector field a p-f vector field, and its flow a polynomial flow. P-f vector fields include, but are not limited to, linear ones. Bass and Meisters (BM) show that if V is a p-f vector field then (1) is complete--solutions are defined for all real t. We show that if V is a p-f vector field then solutions of (1) extend to entire functions of complex t. Call a polynomial map P:R$\sp{n} \to {\bf R}\sp{n}$ a p-symmetry of V if P has a polynomial inverse and maps solutions of (1) to solutions of (1). One question that has not been answered using the results in (BM) is the following: Do the Lorenz equations (GH,Sp) (UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\vbox{\halign{#\hfil\qquad&\hfil#\hfil\cr $\dot x = \sigma(y-x)$& \hfil\cr $\dot y = \rho x-y-xz$& $(x,y,z)\in {\bf R}\sp3$\cr $\dot z = -\beta z + xy$& $\sigma,\rho,\beta > 0$\cr}}$$(TABLE/EQUATION ENDS)have a polynomial flow? By inspecting their p-symmetries we show they do not. Bass and Meisters show that p-f vector fields are polynomial. Therefore assume that V is polynomial. Define $c\sb{k}\:{\bf R}\sp{n}$ $\to$ ${\bf R}\sp{n}$ by $c\sb{k+1}(x)$ = $(Dc\sb{k})(x){\bf V}(x)$, $k \geq 0$, $c\sb0(x)$ = $x$ where D is the derivative operator. We show that there exists a neighborhood U of $\{0\}$ $\times\ {\bf R}\sp{n}$ in R $\times\ {\bf R}\sp{n}$ where the flow $\phi$ can be expressed as $\phi(t,x)$ = $\sum\sbsp{k=0}{\infty}\ {c\sb{k}(x)\over k!}\ t\sp{k},$ $(t,x) \in U$, and V is a p-f vector if and only if the degrees of the components of the $c\sb{k}$ are bounded above (define deg0 = $-\infty$). There is a connection between polynomial flows and an open question from Algebraic Geometry. Meisters and Olech (MO1) show that Keller's Jacobian conjecture in R$\sp{n}$ is true if and only if a certain conjecture, formulated in terms of polynomial flows, is true. We give a complete classification of the p-symmetries of p-f vector fields on R$\sp2$. Then, using this classification, we show that the Jacobian conjecture in R$\sp2$ is true if and only if a certain conjecture, formulated in terms of p-symmetries, is true. In addition to the work on polynomial flows, we present some easily applied sufficient conditions for a differentiable map D:R$\sp{n} \to {\bf R}\sp{n}$ to be one-to-one and give a general principle for formulating more sufficient conditions.
Subject Area
Mathematics
Recommended Citation
Coomes, Brian Arthur, "Polynomial flows, symmetry groups and conditions sufficient for injectivity of maps" (1988). ETD collection for University of Nebraska-Lincoln. AAI8824921.
https://digitalcommons.unl.edu/dissertations/AAI8824921