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Direct sum condition and Artin Approximation in Banach spaces

  • The system of undetermined coefficients of a bifurcation problem ?[?] = 0 in Banach spaces is investigated for proving the existence of families of solution curves by use of the implicit function theorem. The main theorem represents an Artin-Tougeron type result in the sense that approximation of order 2? ensures exact solutions agreeing up to order ? with the approximation [13], [22]. Alternatively, it may be interpreted as Hensel’s Lemma in Banach spaces. In the spirit of [9] and [18], the required surjectivity condition is interpreted as a direct sum condition of order ? that allows for solving the remainder equation with respect to graded subspaces derived from an appropriate filtration [24], [25]. In the direction of these subspaces, the determinant can be calculated in a finite dimensional setting, enabling the investigation of secondary global bifurcation phenomena by sign change of Brouwer’s degree [18]. The direct sum condition seems to be a generalization of the direct sum condition introduced in [9]. The implicitThe system of undetermined coefficients of a bifurcation problem ?[?] = 0 in Banach spaces is investigated for proving the existence of families of solution curves by use of the implicit function theorem. The main theorem represents an Artin-Tougeron type result in the sense that approximation of order 2? ensures exact solutions agreeing up to order ? with the approximation [13], [22]. Alternatively, it may be interpreted as Hensel’s Lemma in Banach spaces. In the spirit of [9] and [18], the required surjectivity condition is interpreted as a direct sum condition of order ? that allows for solving the remainder equation with respect to graded subspaces derived from an appropriate filtration [24], [25]. In the direction of these subspaces, the determinant can be calculated in a finite dimensional setting, enabling the investigation of secondary global bifurcation phenomena by sign change of Brouwer’s degree [18]. The direct sum condition seems to be a generalization of the direct sum condition introduced in [9]. The implicit function theorem delivers stability of ? leading coefficients [?̅1 , … , ??̅] with respect to perturbations of order 2? +1 and uniqueness in pointed wedges around the solution curves. Further, denoting by ?̅∞ the subset of arc space ?∞ with [?̅1 , … , ??̅] fixed, ?̅∞ is iteratively approximated by ?̅ 2?+? as ? → ∞. In addition, a lower bound of the Greenberg function ?(?) of a singularity is constructed by use of a step function obtained from ?-degree of different solution curves. Finally, based on Kouchnirenko’s theorem [17] and an extension in [6], the results are applied to Newton-polygons where it is shown that the Milnor number of a singularity can be calculated by the sum of ?-degrees of corresponding solution curves. Simple ???-singularities are investigated in detail. The main theorem represents a version of strong implicit function theorem in Banach spaces, possibly comparable to theorems in [2]. Moreover, our aim is to extend the direct sum condition of order ? from [9] to certain topics in singularity and approximation theory. Generalizations to modules as well as specializations to polynomials seem to be promising.show moreshow less

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Metadaten
Author:Matthias StiefenhoferORCiD
URN:urn:nbn:de:bvb:859-5835
Place of publication:Kempten
Document Type:Preprint
Language:English
Date of first Publication:2019/07/20
Publishing Institution:Hochschule für angewandte Wissenschaften Kempten
Tag:AlgGeo; Artin approximation; Greenberg function; Milnor number; arc space
Number of pages:38 Seiten
Original Publication / Source:Stiefenhofer, Matthias (2019): Direct sum condition and Artin Approximation in Banach spaces, arXiv.org, New York, Cornell University
https://arxiv.org/abs/1905.07583,
https://doi.org/10.48550/arXiv.1905.07583
Institutes:Fakultät Maschinenbau
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik
Open Access:open_access
Publication Lists:Stiefenhofer, Matthias
Publication reviewed:nicht begutachtet
Licence (German):Creative Commons - CC BY-NC-SA - Namensnennung - Nicht kommerziell - Weitergabe unter gleichen Bedingungen 4.0 International
Release Date:2021/02/15
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