Solutions of quasianalytic equations
| Autor: | André Belotto da Silva, Iwo Biborski, Edward Bierstone |
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| Rok vydání: | 2017 |
| Předmět: |
Class (set theory)
Pure mathematics Formal power series Mathematics::Complex Variables General Mathematics 010102 general mathematics Mathematics::Classical Analysis and ODEs Structure (category theory) General Physics and Astronomy Function (mathematics) 01 natural sciences symbols.namesake Factorization 0103 physical sciences Taylor series symbols 010307 mathematical physics 0101 mathematics Equation solving Mathematics |
| Zdroj: | Selecta Mathematica. 23:2523-2552 |
| ISSN: | 1420-9020 1022-1824 |
| DOI: | 10.1007/s00029-017-0345-3 |
| Popis: | The article develops techniques for solving equations $$G(x,y)=0$$ , where $$G(x,y)=G(x_1,\ldots ,x_n,y)$$ is a function in a given quasianalytic class (for example, a quasianalytic Denjoy–Carleman class, or the class of $${\mathcal C}^\infty $$ functions definable in a polynomially-bounded o-minimal structure). We show that, if $$G(x,y)=0$$ has a formal power series solution $$y=H(x)$$ at some point a, then H is the Taylor expansion at a of a quasianalytic solution $$y=h(x)$$ , where h(x) is allowed to have a certain controlled loss of regularity, depending on G. Several important questions on quasianalytic functions, concerning division, factorization, Weierstrass preparation, etc., fall into the framework of this problem (or are closely related), and are also discussed. |
| Databáze: | OpenAIRE |
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