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

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

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]**H. Bass and G. H. Meisters,*Polynomial flows in the plane*, Adv. in Math.**55**(1985), 173-208. MR**772614 (86c:58127)****[2]**B. Coomes,*Polynomial flows, symmetry groups, and conditions sufficient for injectivity of maps*, doctoral thesis, University of Nebraska, 1988.**[3]**-,*The Lorenz system does not have a polynomial flow*, J. Differential Equations, (to appear). MR**1027976 (91b:58213)****[4]**-,*Polynomial flows on*, (to appear).**[5]**S. Mandelbrojt, Analytic functions and classes of infinitely differentiable functions, The Rice Institute Phamphlet, Vol. XXIX, no. 1, pp. 2-3, January 1942. MR**0006354 (3:292d)****[6]**G. H. Meisters,*Jacobian problems in differential equations and algebraic geometry*, Rocky Mountain J. Math.**12**(1982), 679-705. MR**683862 (84c:58048)****[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]**G. H. Meisters and C. Olech,*A poly-flow formulation of the Jacobian conjecture*, Bull. of the Polish Academy of Sciences Mathematics**35**(1987), pp. 725-731. MR**961711 (89j:13005)**

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Additional Information

DOI:
https://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