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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.
Equations of the form ?[?] = 0, ?: ? ⟶ ?̅ smooth, ?, ?̅ real Banach spaces are investigated with the aim of continuing the basic solution ?[0] = 0 with ? ′ [0] Fredholm operator to a solution curve ?[?(?)] = 0 with the implicit function theorem. If ? ′ [0] is surjective, then the transversality condition of the implicit function theorem can be satisfied in a straightforward way, yielding a regular solution curve, whereas otherwise the equation ?[?] = 0 has to be extended appropriately for reaching a surjective linearization accessible to the implicit function theorem. This extension process, implying in the first step the standard bifurcation theorem of simple bifurcation points, is continued arbitrarily, yielding a sequence of bifurcation results presumably being applicable to bifurcation points with finite degeneracy.
A note on an expansion formula with application to nonlinear Differential-Algebraic-Equations
(2019)
In [DL] systems of differential polynomials are investigated with
respect to properties of Artin approximation type. The key tool in [DL] is an extended version of a formula by Hurwitz [Hu] expressing high order derivatives of an expansion by lower ones. The formula is further refined in [VFZ] to deliver sufficient conditions concerning the existence of power series solutions of scalar algebraic differential equations of order?.
In the paper at hand, the main results from [VFZ] are first reproduced and further extended to systems of nonlinear differential algebraic equations. In addition, a simple extension of Tougeron’s implicit function theorem is given in a specific constellation.
The results follow from [S1], [S2] where Artin approximation is treated within a Banach space setting, thereby constructing an expansion formula that expresses accurately the required dependency of low and high order derivatives within the system of undetermined coefficients.
We study the action of the nonlinear mapping G[z] between real or complex Banach spaces in the vicinity of a given curve with respect to possible linearization, emerging patterns of level sets, as well as existing solutions of G[z]=0. The results represent local generalizations of the standard implicit or inverse function theorem and of Newton's Lemma, considering the order of approximation needed to obtain solutions of G[z]=0.
The main technical tool is given by Jordan chains with increasing rank, used to obtain an Ansatz, appropriate for transformation of the nonlinear system to its linear part. The family of linear mappings is restricted to the case of an isolated singularity.
Geometrically, the Jordan chains define a generalized cone around the given curve, composed of approximate solutions of order 2k with k denoting the maximal rank of Jordan chains needed to ensure k-surjectivity of the linear family. Along these lines, the zero set of G[z] in the cone is calculated immediately, agreeing up to the order of k−1 with the given approximation. Hence, the results may also be interpreted as a version of Tougeron's implicit function theorem or Hensel's Lemma in Banach spaces, essentially restricted to the arc case of a single variable.
Finally, by considering a left shift of the Jordan chains, the Ansatz can be modified in a systematic way to obtain a sequence of refined versions of linearization theorems and Newton Lemmas in Banach spaces.
Tougeron’s implicit function theorem and Hensel’s lemma are well known representatives concerning 2?-approximation/?-nondegeneracy implying existence of solutions with identity of order ?. This note aims to extend this
principle to equations ?[?] = 0 in Banach spaces, using ?-transversality concepts, which may geometrically be interpreted as generalized cones spanned by
submanifolds, each characterized by a certain expansion rate.
The number of manifolds in the cone, as well as their expansion rates, is recursively increased until an appropriate desingularization of the cone is build up
with linearization expressed by first ? + 1 derivatives of the singular operator
at the base point.
Along these lines, a well-defined submersion is constructed in the cone with
uniformly bounded inverse when approaching the singularity. The techniques
are restricted to curves, possibly touching by high order the singular locus of ?, but ultimately traversing it, in this way defining an isolated singularity of the
operator family given by the linearization along the curve.
The fine resolution of the cone by the manifolds represents an improvement
compared to measuring the variation of the nonlinear operator exclusively by
the overall behaviour of the determinant.
In case of finite dimensions, each half-cone is characterized by a constant topological degree that can be used to investigate a solution curve in general position with respect to secondary bifurcation.
The core of all considerations is given by some characteristic patterns, valid in
the system of undetermined coefficients that allow for detailed analysis of the
power series resulting from plugging the power series of the ansatz into the
power series of the nonlinear operator.
We give conditions for local diagonalization of analytic operator families acting between real or complex Banach spaces.The transformations are constructed from an operator Töplitz matrix obtained from Jordan chains of increasing length. The basic assumption is given by stabilization of the Jordan chains at length k in the sense that no root elements with finite rank above k are allowed to exist. Jordan chains with infinite rank may appear. These assumptions ensure finite pole order equal to k of the generalized inverse. The Smith form arises immediately.
Smooth continuation of kernels and ranges towards appropriate limit spaces is considered using associated families of analytic projection functions.
No Fredholm properties or other finiteness assumptions, besides the pole order, are assumed. Real and complex Banach spaces are treated without difference by elementary analysis of the system of undetermined coefficients.
Formal power series solutions of the system of undetermined coefficients are constructed, which are turning into convergent solutions, as soon as analyticity of the operator family and continuity of the projections is assumed. Along these lines, results concerning linear Artin approximation follow immediately, which are well known in finite dimensions. The main technical tool is given by a defining equation of Nakayama Lemma type.