Germs Of Functions Research Articles

Characterization of plane to plane map germs by invariants

We characterize the map germs of corank at most 11 with AA-codimension ≤6≤6 in terms of certain invariants such as multiplicity and the number of cusps of map germs. On the basis of this characterization we present an algorithm to classify the map germs of corank at most 11 from (C2,0)(C2,0) to (C2,0)(C2,0) with AA-codimension ≤6≤6 and also give its implementation in the computer algebra system SINGULAR.

Vanishing cycle control by the lowest degree stalk cohomology

Given the germ of an analytic function on affine space with a smooth critical locus, we prove that the constancy of the reduced cohomology of the Milnor fiber in lowest possible non-trivial degree off a codimension two subset of the critical locus implies that the vanishing cycles are concentrated in lowest degree and are constant.

The dual tree of a fold map germ from to

Let $f\colon (\mathbb {R}^{3},0)\to (\mathbb {R}^{4},0)$ be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of $f$ with a small enough sphere $S^{3}_\epsilon$ centred at the origin in $\mathbb {R}^{4}$. If $f$ is of fold type, we define a tree, that we call dual tree, that contains all the topological information of the link and we prove that in this case it is a complete topological invariant. As an application we give a procedure to obtain normal forms for any topological class of fold type.

Wave maps and constant curvature surfaces: singularities and bifurcations

Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructing germs of Lorentzian-harmonic maps from their $k$-jets and use this construction to study the singularities of such maps. We also show how to construct pseudospherical surfaces with prescribed singularities using loop groups. We study the singularities of pseudospherical surfaces and obtain their bifurcations in generic 1-parameter families of such surfaces.

Cohomology of contact loci

We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean’s spectral sequence converging to the Floer cohomology of the $m$‑th iterate of the monodromy, when the polynomial has an isolated singularity. Inspired by this connection, we conjecture that if two germs of holomorphic functions are embedded topologically equivalent, then the Milnor fibers of their tangent cones are homotopy equivalent.

BI-LIPSCHITZ CONTACT INVARIANCE OF RANK

We address the question of whether the Thom-Boardman symbol of a map germ is an invariant with respect to bi-Lipschitz right equivalence. We give a counterexample showing that in general the answer is negative. We prove that the rank of a map germ is a bi-Lipschitz contact invariant. Consequently, the first Thom-Boardman symbol and its length are bi-Lipschitz contact invariants.

TOPOLOGICAL INVARIANTS AND MILNOR FIBRE FOR \(\mathcal{A}\)-FINITE GERMS \(C^2\) to \(C^3\)

This note is the observation that a simple combination of known results shows that the usual analytic invariants of a finitely determined multi-germ \(f : (C^2 , S) → (C^3 , 0) \)—namely, the image Milnor number , the number of cross-caps and triple points, \(C\) and \(T\), and the Milnor number \(μ(Σ)\) of the curve of double points in the target—depend only on the embedded topological type of the image of \(f\). As a consequence, one obtains the topological invariance of the sign-refined Smale invariant for immersions \(j : S^3 \looparrowright S^5\) associated to finitely determined map germs \((C^2 , 0) → (C^3 , 0)\).

A generalization of Zakalyukin's lemma, and symmetries of surface singularities

Zakalyukin's lemma asserts that the coincidence of the images of two wave front germs implies the right equivalence of corresponding map germs under a certain genericity assumption. The purpose of this paper is to give an improvement of this lemma for frontals. Moreover, we give several applications for singularities on surfaces.

Homologically trivial integrable deformations of germs of holomorphic functions

Homologically trivial integrable deformations of germs of holomorphic functions

Clustering polar curves

This essay builds on the idea of grouping the polar curves of 2-variable function germs in polar clusters. One first obtains a bijective correspondence between certain partitions of the set of polar quotients of two topologically equivalent function germs. We explain how this bijective correspondence may be refined in the Lipschitz category in terms of the associated gradient canyons.

Pluriharmonic Definable Functions in Some O-Minimal Expansions of the Real Field

In this paper, we first try to solve the following problem: If a pluriharmonic function $f$ is definable in an arbitrary o-minimal expansion of the structure of the real field $\overline{\mathbb{R}}:=(\mathbb{R},+,-,.,0,1,<)$, does this function be locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of $\mathbb{R}^n$. We obtain the same answer for an o-minimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at $0\in\mathbb{R}^n$ which are definable in the o-minimal structure $(\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})$, here $\alpha_1,\ldots,\alpha_p$ are irrational real numbers.

The polynomial Hermite-Padé -system for meromorphic functions on a compact Riemann surface

Given a tuple of germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Padé -system, which includes the Hermite-Padé polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Padé -system constructed from the tuple of germs of functions that are meromorphic on an -sheeted compact Riemann surface . We show that if for some meromorphic function on , then with the help of the ratios of polynomials of the Hermite-Padé -system we recover the values of on all sheets of the Nuttall partition of , apart from the last sheet. Bibliography: 18 titles.

Patching over analytic fibers and the local–global principle

As a starting point for higher-dimensional patching in the Berkovich setting, we show that this technique is applicable around certain fibers of a relative Berkovich analytic curve. As a consequence, we prove a local–global principle over the field of overconvergent meromorphic functions on said fibers. By showing that these germs of meromorphic functions are algebraic, we also obtain local–global principles over function fields of algebraic curves defined over a class of (not necessarily complete) ultrametric fields, thus generalizing the results of Mehmeti(Compos Math 155:2399–2438, 2019).

Choroidal metastasis as initial presentation of aggressive medullary thyroid carcinoma with widespread mediastinal, brain, pituitary, bone, lung, and liver metastasis: Case report and literature review.

IntroductionMedullary thyroid carcinoma (MTC) is a neuroendocrine tumor that originates from the parafollicular C cells of the thyroid gland. MTC can be due to sporadic or hereditary causes due to gain of function germ line mutations in the RET proto-oncogene. MTC presenting as ocular symptoms due to choroidal mass is rare with bad prognosis.Presentation of caseA 38-year-old Sudanese male presented to Hamad General Hospital, complaining of sudden painless decrease of vision of the right eye of 3 weeks duration. After investigations using imaging methods, the patient was discovered to have metastatic MTC that presented as choroidal mass and metastasized to his lung, bone, brain, pituitary, liver and mediastinum.DiscussionIn terms of investigations, serum levels of calcitonin have superior diagnostic accuracy. Our patient undertook diagnostic imaging including ultrasonography, fine needle aspiration and computerized tomography (CT) scan and/or MRI imaging. He undertook total thyroidectomy and left neck dissection followed by stereotactic radiosurgery for the right orbit and pituitary. He then received systemic anti-RET therapy (Selpercatinib). At 5 months follow up there was dramatic drop in CEA from 888 μg/L to 164 μg/L, and calcitonin from >585.2 pmol/L to 354 pmol/L.ConclusionChoroidal metastasis as initial presentation of MTC is extremely rare and challenging to diagnose. Surgeons need a high index of suspicion when ocular symptoms accompany a neck mass or thyroid-related symptoms. MTC has a progressive course with involvement of blood vessels and neck lymph nodes. Choroidal metastasis of MTC is challenging to manage.

A proof of A. Gabrielov’s rank theorem

This article contains a complete proof of Gabrielov’s rank theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung theorem. We include, furthermore, new extensions of the rank theorem (concerning the Zariski main theorem and elimination theory) to commutative algebra.

Uniform stable radius and Milnor number for non-degenerate isolated complete intersection singularities

We prove that for two germs of analytic mappings $$f,g:({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}}^p,0)$$ with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family $$\{f_t\}$$ of analytic maps with $$f_0=f, f_1=g$$ which has a so-called uniform stable radius for the Milnor fibration. As a corollary, we show that their Milnor numbers are equal. Also, a formula for the Milnor number is given in terms of the Newton polyhedra of the component functions. This is a generalization of the result by C. Bivia-Ausina. Consequently, we obtain that the Milnor number of a non-degenerate isolated complete intersection singularity is an invariant of Newton boundaries.

Triangular resolutions and effectiveness for holomorphic subelliptic multipliers

A solution to the effectiveness problem in Kohn's algorithm for generating holomorphic subelliptic multipliers is provided for general classes of domains of finite type in Cn, that include the so-called special domains given by finite and infinite sums of squares of absolute values of holomorphic functions. Also included is a more general class of domains recently discovered by M. Fassina [23]. More generally, for any smoothly bounded pseudoconvex domain we introduce an invariantly defined associated sheaf S of C-subalgebras of holomorphic function germs, that combined with a result of Fassina, reduces the existence of effective subelliptic estimates at p to a purely algebraic geometric question of controlling the multiplicity of S.Our main new tool, a triangular resolution, is the construction of subelliptic multipliers decomposable as Q∘Γ, where Γ is constructed from pre-multipliers and Q is part of a triangular system. The effectiveness is proved via a sequence of newly proposed procedures, called here meta-procedures, built on top of the holomorphic Kohn's procedures, where the order of subellipticity can be effectively tracked. Important sources of inspiration are algebraic geometric techniques by Y.-T. Siu [54,55] and procedures for triangular systems by D.W. Catlin and J.P. D'Angelo [16,8].The proposed procedures are purely algebraic and as such can be of wider interest for geometric and computational problems involving Jacobian determinants, such as resolving singularities of holomorphic maps.

Holonomy and Equivalence of Analytic Foliations

The main goal of this paper is the analytic classification of the germs of singular foliations generated, up to an analytic change of coordinates, by the germs of vector fields of form the \(x\partial _{x}+{\sum }_{i=1}^{n}a_{i}(x,\mathbf {z})\partial _{z_{i}}\), where ai(x,z) is a germ of analytic function with ai(x,0) = 0. We prove, under some hypothesis, that these germs of singular foliations are analytically classified once their local holonomy along a given separatrix is analytically conjugated.

An Implementation of Lipschitz Simple Functions in Computer Algebra System Singular

A complete classification of simple function germs with respect to Lipschitz equivalence over the field of complex numbersℂwas given by Nguyen et al. The aim of this article is to implement a classifier in terms of easy computable invariants to compute the type of the Lipschitz simple function germs without computing the normal form in the computer algebra system Singular.

The local image problem for complex analytic maps

We address the question "when the local image of a map is well defined" and answer it in case of holomorphic map germs with target $(\bC^{2}, 0)$. We prove a criterion for holomorphic map germs $(X, x)\to (Y, y)$ to be locally open, solving a conjecture by Huckleberry in all dimensions.