Smooth polynomial paths with nonanalytic tangents

Abstract:

We prove that there exist ${C^\infty }$ functions $\varphi :{{\mathbf {R}}_t} \times {{\mathbf {R}}_x} \to {\mathbf {R}}$ such that although $\varphi \left ( {t,x} \right )$ is a polynomial in $x$ for each $t$ in ${\mathbf {R}},\dot \varphi \left ( {0,x} \right ) \equiv \left ( {\partial \varphi /\partial t} \right )\left ( {0,x} \right )$ need not even be analytic in $x$ let alone polynomial. It was shown earlier by one of the authors [Meisters] that this cannot happen if $\varphi$ satisfies the group-property (even locally) of flows, namely if $\varphi \left ( {s,\varphi \left ( {t,x} \right )} \right ) = \varphi \left ( {s + t,x} \right )$ .