Polynomial flows, symmetry groups and conditions sufficient for injectivity of maps

Consider the initial value problem(UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}dot y (equiv {lcub}dyover dt{rcub}) = {lcub}bf V{rcub}(y), y(0) = x in {lcub}bf R{rcub}sp{lcub}n{rcub}eqno(1){dollar}{dollar}(TABLE/EQUATION ENDS)where V is a {dollar}Csp1{dollar} vector field on R{dollar}sp{lcub}n{rcub}{dollar} and for each t the solution (flow) {dollar}phi{dollar} 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{dollar}sp{lcub}n{rcub} to {lcub}bf R{rcub}sp{lcub}n{rcub}{dollar} 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){dollar}{dollar}vbox{lcub}halign{lcub}#hfilqquad&hfil#hfilcr {dollar}dot x = sigma(y-x){dollar}& hfilcr {dollar}dot y = rho x-y-xz{dollar}& {dollar}(x,y,z)in {lcub}bf R{rcub}sp3{dollar}cr {dollar}dot z = -beta z + xy{dollar}& {dollar}sigma,rho,beta > 0{dollar}cr{rcub}{rcub}{dollar}{dollar}(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 {dollar}csb{lcub}k{rcub}:{lcub}bf R{rcub}sp{lcub}n{rcub}{dollar} {dollar}to{dollar} {dollar}{lcub}bf R{rcub}sp{lcub}n{rcub}{dollar} by {dollar}csb{lcub}k+1{rcub}(x){dollar} = {dollar}(Dcsb{lcub}k{rcub})(x){lcub}bf V{rcub}(x){dollar}, {dollar}k geq 0{dollar}, {dollar}csb0(x){dollar} = {dollar}x{dollar} where D is the derivative operator. We show that there exists a neighborhood U of {dollar}{lcub}0{rcub}{dollar} {dollar}times {lcub}bf R{rcub}sp{lcub}n{rcub}{dollar} in R {dollar}times {lcub}bf R{rcub}sp{lcub}n{rcub}{dollar} where the flow {dollar}phi{dollar} can be expressed as {dollar}phi(t,x){dollar} = {dollar}sumsbsp{lcub}k=0{rcub}{lcub}infty{rcub} {lcub}csb{lcub}k{rcub}(x)over k!{rcub} tsp{lcub}k{rcub},{dollar} {dollar}(t,x) in U{dollar}, and V is a p-f vector if and only if the degrees of the components of the {dollar}csb{lcub}k{rcub}{dollar} are bounded above (define deg0 = {dollar}-infty{dollar}).; 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{dollar}sp{lcub}n{rcub}{dollar} 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{dollar}sp2{dollar}. Then, using this classification, we show that the Jacobian conjecture in R{dollar}sp2{dollar} 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{dollar}sp{lcub}n{rcub} to {lcub}bf R{rcub}sp{lcub}n{rcub}{dollar} to be one-to-one and give a general principle for formulating more sufficient conditions.