Implicit function theorem and codimension estimation relative to ${\bm t}$-${\bm P}$-${\mathcal{K}}$-equivalence
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摘要: 证得光滑映射芽在\,${t}$-${P}$-${\mathcal{K}}$-等价关系下的隐函数定理, 并对其余维进行估计. 所得结果可为对具有区别参数的光滑映射芽的分类研究提供有力工具, 也可成为讨论完全可积微分方程芽分支的基础.
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关键词:
- 光滑映射芽 /
- $t$-$P$-${\mathcal{K}}$-等价 /
- 隐函数定理 /
- 余维
Abstract: The implicit function theorem under $t$-$P$-${\mathcal{K}}$-equivalence relation was obtained, and the $t$-$P$-${\mathcal{K}}$-codimension was also estimated. The result may be a powerful tool for the classification of smooth mapping germs with distinguished parameters relative to $t$-$P$-${\mathcal{K}}$-equivalence, and for the study of bifurcations of differential equation germs with complete integral. -
[1] {1}MATHER J N. Stability of\,$C^{\infty}$\,mappings, Ⅰ: The divisiontheorem[J]. Ann of Math, 1968, 87(2): 89--104.{2}MATHER J N. Stability of\,$C^{\infty}$\,mappings, Ⅱ: Infinitesimalstability implies stability[J]. Ann of Math, 1969, 89(2): 254-291.{3}MATHER J N. Stability of\,$C^{\infty}$\,mappings, Ⅲ: Finitelydetermined map germs[J]. Publ Math IHES, 1968: 279-308.{4}MATHER J N. Stability of\,$C^{\infty}$\,mappings, Ⅳ:Classification of stable germs by R-algebras[J]. Publ Math IHES,1969: 223--248.{5}MATHER J N. Stability of mappings, Ⅴ: Transversality[J]. AdvancesIn Mathematics, 1970: 301-336.{6}MATHER J N. Stability of mappings, Ⅵ: The nice dimensions[J].Proc of Liverpool Symposium I, 1970: 207-253.{7}BIRBRAIR L, COSTA J C F, FERNANDES A. Finiteness theorem fortopological contact equivalence of map germs[J]. HokkaidoMathematical Journal, 2009, 38: 511-517.{8}IZUMIYA S. Perestroikas of optical wave fronts and graphlikeLegendrian unfoldings[J]. J Diff Geom, 1993, 38: 485-500.{9}ZAKALYUKIN V M. Reconstruction of fronts and caustics depending on aparameter and versality of mappings[J]. J of Sovient Math, 1984, 27:2713-2735.{10}TAKAHASHI M. Bifurcations of completely integrable first-orderordinary differential equations[J]. Journal of MathematicalSciences. 2007, 144(1): 3854--3869.{11}TSUKADA T. A generic classification of function germs with respectto the reticular equivalence[J]. Hokkaido Mathematical Journal,2009, 38: 177-203.{12}IZUMIYA S. Generic bifurcations of varieties[J]. Manuscripta Math,1984, 46: 137-164.
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