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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Smooth polynomial paths with nonanalytic tangents


Authors: Robert M. McLeod and Gary H. Meisters
Journal: Proc. Amer. Math. Soc. 107 (1989), 697-700
MSC: Primary 26E10; Secondary 14E07, 58C27
MathSciNet review: 987612
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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)$ .


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0987612-7
PII: S 0002-9939(1989)0987612-7
Keywords: Polyomorphism, polynomial flows, polynomial vector field, smooth $ ({C^\infty })$ polynomial path, nonanalytic tangent, tangent to path in polynomial space
Article copyright: © Copyright 1989 American Mathematical Society