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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Determinacy of sufficiently differentiable maps


Author: Alan M. Selby
Journal: Trans. Amer. Math. Soc. 312 (1989), 85-113
MSC: Primary 58C27
DOI: https://doi.org/10.1090/S0002-9947-1989-0937251-3
MathSciNet review: 937251
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Abstract: Variants of the algebraic conditions of Mather are shown to be sufficient for the $ k$-determinacy of $ {C^u}$ maps with respect to $ j$-flat, contact (or right) $ {C^r}$ equivalence relations where $ u - k \leq r \leq u - k + j + 1$ and $ 0 \leq j < k \leq u$. The required changes of coordinates and matrix-valued functions are constructed from the variation of coefficients in polynomials. The main result follows from a finite-dimensional, polynomial pertubation argument which employs a parameter-dependent polynomial representation of functions based on Taylor's formula. For $ r > k$, the algebraic conditions are seen to be necessary.


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DOI: https://doi.org/10.1090/S0002-9947-1989-0937251-3
Keywords: Algebraic conditions for $ {C^r}$ contact and right determinacy, Taylor's formula, polynomial representation of maps, ordinary differential equations
Article copyright: © Copyright 1989 American Mathematical Society