TY - INPR A1 - Stiefenhofer, Matthias T1 - Direct sum condition and Artin Approximation in Banach spaces N2 - 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 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. KW - AlgGeo KW - Artin approximation KW - arc space KW - Greenberg function KW - Milnor number Y1 - 2019 UR - https://opus4.kobv.de/opus4-hs-kempten/frontdoor/index/index/docId/583 UR - https://nbn-resolving.org/urn:nbn:de:bvb:859-5835 CY - Kempten ER -