American Mathematical Society, Proceedings of the American Mathematical Society, 3(107), p. 697-700, 1989
DOI: 10.1090/s0002-9939-1989-0987612-7
American Mathematical Society, Proceedings of the American Mathematical Society, 3(107), p. 697
DOI: 10.2307/2048167
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We prove that there exist C ∞ {C^∞ } functions φ : R t × R x → R φ :{{\mathbf {R}}_t} \times {{\mathbf {R}}_x} \to {\mathbf {R}} such that although φ ( t , x ) φ \left ( {t,x} \right ) is a polynomial in x x for each t t in R , φ ˙ ( 0 , x ) ≡ ( ∂ φ / ∂ t ) ( 0 , x ) {\mathbf {R}},\dot φ \left ( {0,x} \right ) ≡ \left ( {𝟃 φ /𝟃 t} \right )\left ( {0,x} \right ) need not even be analytic in x x let alone polynomial. It was shown earlier by one of the authors [Meisters] that this cannot happen if φ φ satisfies the group-property (even locally) of flows, namely if φ ( s , φ ( t , x ) ) = φ ( s + t , x ) φ \left ( {s,φ \left ( {t,x} \right )} \right ) = φ \left ( {s + t,x} \right ) .