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 | References | Similar Articles | Additional Information

Abstract: We prove that there exist functions such that although is a polynomial in for each in need not even be analytic in let alone polynomial. It was shown earlier by one of the authors [Meisters] that this cannot happen if satisfies the group-property (even locally) of flows, namely if .

**[1]**Hyman Bass and Gary Meisters,*Polynomial flows in the plane*, Adv. in Math.**55**(1985), no. 2, 173–208. MR**772614**, 10.1016/0001-8708(85)90020-9**[2]**B. Coomes,*Polynomial flows, symmetry groups, and conditions sufficient for injectivity of maps*, doctoral thesis, University of Nebraska, 1988.**[3]**Brian A. Coomes,*The Lorenz system does not have a polynomial flow*, J. Differential Equations**82**(1989), no. 2, 386–407. MR**1027976**, 10.1016/0022-0396(89)90140-X**[4]**-,*Polynomial flows on*, (to appear).**[5]**S. Mandelbrojt,*Analytic functions and classes of infinitely differentiable functions*, Rice Inst. Pamphlet**29**(1942), no. 1, 142. MR**0006354****[6]**Gary H. Meisters,*Jacobian problems in differential equations and algebraic geometry*, Rocky Mountain J. Math.**12**(1982), no. 4, 679–705. MR**683862**, 10.1216/RMJ-1982-12-4-679**[7]**-,*Polynomial flows on*, Banach Center Publications (Volume on the Dynamical Systems Semester held at the Stefan Banach International Mathematical Center, . Mokotowska 25, Warszawa Poland, Autumn 1986), (to appear).**[8]**Gary H. Meisters and Czesław Olech,*A poly-flow formulation of the Jacobian conjecture*, Bull. Polish Acad. Sci. Math.**35**(1987), no. 11-12, 725–731 (English, with Russian summary). MR**961711**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1989-0987612-7

Keywords:
Polyomorphism,
polynomial flows,
polynomial vector field,
smooth polynomial path,
nonanalytic tangent,
tangent to path in polynomial space

Article copyright:
© Copyright 1989
American Mathematical Society