Germs Of Functions Research Articles

Local systems on analytic germ complements

We prove that the cohomology jump loci of rank one local systems on the complement in a small ball of a germ of a complex analytic set are finite unions of torsion translates of subtori. This is a generalization of the classical Monodromy Theorem stating that the eigenvalues of the monodromy on the cohomology of the Milnor fiber of a germ of a holomorphic function are roots of unity.

The Reeb Graph of a Map Germ from ℝ3 to ℝ2 with Isolated Zeros

AbstractWe consider finitely determined map germs f : (ℝ3, 0) → (ℝ2, 0) with f–1(0) = {0} and we look at the classification of this kind of germ with respect to topological equivalence. By Fukuda's cone structure theorem, the topological type of f can be determined by the topological type of its associated link, which is a stable map from S2 to S1. We define a generalized version of the Reeb graph for stable maps γ : S2→ S1, which turns out to be a complete topological invariant. If f has corank 1, then f can be seen as a stabilization of a function h0: (ℝ2, 0) → (ℝ, 0), and we show that the Reeb graph is the sum of the partial trees of the positive and negative stabilizations of h0. Finally, we apply this to give a complete topological description of all map germs with Boardman symbol Σ2, 1.

On the number of topological orbits of complex germs in classes (xy, xa + yb)

We show that there exist an infinite number of topological orbits in classes of complex map germs from the plane to the plane that have a representative of type (xy, xa + yb), with (a, b) ≠ = (2, 3) or (2, 5). Our key tool to prove this existence is the existence (or not) of stems in the class; these germs are not -finitely determined and allow the determination of a non-finite number of topological orbits. We also show that the class (xy, x2 + y5) has two topological orbits.

Topological triviality of families of map germs from $\mathbb R^3$ to $\mathbb R^3$

We show that a one-parameter unfolding $ F:(\mathbb {R}^3 \times \mathbb {R}, 0) \rightarrow (\mathbb {R}^3 \times \mathbb {R}, 0) $ of a finitely determined map germ $f$, with $S(f)$ regular, is topologically trivial if it is excellent in the sense of Gaffney, and the family of the double point curves and cuspidal edges $D(f_t) \cup C(f_t)$ is topologically trivial.

Holomorphic Deformations of Real-Analytic CR Maps and Analytic Regularity of CR Mappings

Let \(M\subset {\mathbb {C}}^N\) and \(M'\subset {\mathbb {C}}^{N'}\) be real-analytic CR submanifolds, with M minimal. We provide a new sufficient condition, that happens to be also essentially necessary, for all sufficiently smooth CR maps \(h:U\rightarrow M'\) defined on a connected open subset of M and of rank larger than a prescribed integer r to be real-analytic on a dense open subset of U. This condition corresponds to the nonexistence of nontrivial holomorphic deformations of germs of real-analytic CR mappings whose rank is larger than r. As a consequence, we obtain several new results about analyticity of CR mappings that, at the same time, generalize and unify a number of previous existing ones.

ON THE CLASSIFICATION OF FUNCTION GERMS OF TWO VARIABLES THAT ARE EQUIVARIANT SIMPLE WITH RESPECT TO AN ACTION OF THE CYCLIC GROUP OF ORDER THREE

We consider the problem to classify function germs(C2, 0) (C, 0), that are equivariant simple with respect to nontrivial actions of the group Z3on C2and on C up to equivariant automorphism germs (C2, 0) (C2, 0).The complete classification of such germs is obtained in the case of nonscalar action of Z3on C2that is nontrivial in both coordinates. Namely, a germ is equivariant simple with respect to such a pair of actions if and only if it is equivalent to ine of the following germs:
 
 (x, y) x3k+1+ y2, k 1;
 (x, y) x2y + y3k1, k 2;
 (x, y) x4+ xy3
 (x, y) x4+ y5.

The Lê-Greuel formula for functions on analytic spaces

In this article we give an extension of the Lê-Greuel formula to the general setting of function germs $(f,g)$ defined on a complex analytic variety $X$ with arbitrary singular set, where $f = (f_1,\ldots,f_k): (X,\underline{0}) \to (\mathbb{C}^k,\underline{0})$ is generically a submersion with respect to some Whitney stratification on $X$. We assume further that the dimension of the zero set $V(f)$ is larger than 0, that $f$ has the Thom $a_f$-property with respect to this stratification, and $g: (X,\underline{0}) \to (\mathbb{C},0)$ has an isolated critical point in the stratified sense, both on $X$ and on $V(f)$.

Minimal Whitney stratification and Euler obstruction of discriminants

In this work we show how to obtain the minimal Whitney stratification of the discriminant of finitely determined map germs from \((\mathbb {C}^m,0)\) to \((\mathbb {C}^p,0)\), of corank one if \(n<p\), and only with \(A_k\) singularities, when \(m = n+p\) with \( n \ge 0\). We apply the theory developed by Gaffney which shows how to compute a Whitney stratification of discriminants of any finitely determined holomorphic map germ in the nice dimensions of Mather, or in its boundary. For the pairs cited above we show that both stratifications coincide. We also compute the local Euler obstruction at 0 in a class of discriminants of finitely determined map germs from \(\mathbb {C}^{n+p}\) to \(\mathbb {C}^p\) with \(n\ge 0\) and only with \(A_k\) singularities.

Q-Borel-Laplace summation for q–difference equations with two slopes

In this paper, we consider linear q-difference systems with coefficients that are germs of meromorphic functions, with Newton polygon that has two slopes. Then, we explain how to compute similar meromorphic gauge transformations than those appearing in the work of Bugeaud, using a q-analogue of the Borel–Laplace summation.

Local topological algebraicity of analytic function germs

Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or complex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety.

On Segre Numbers of Homogeneous Map Germs

Segre numbers and Segre cycles of ideals were independently introduced by Tworzewski, by Achilles and Manaresi and by Gaffney and Gassler. They are generalization of the Lê numbers and Lê cycles, introduced by Massey. In this article we give Lê-Iomdine type formulas for these cycles and numbers of arbitrary ideals. As a consequence we give a Plücker type formula for the Segre numbers of ideals generated by weighted homogeneous functions, in terms of their weights and degree. As an application of these results, we compute, in a purely combinatorial manner, the Segre numbers of the ideal which defines the critical loci of a map germ defined by a sequence of central hyperplane arrangements in $\mathsf{C}^{n+1}$.

Quasicircles as equipotential lines, homotopy classes and geodesics

We give an application of our earlier results concerning the quasiconformal extension of a germ of a conformal map to establish that in two dimensions the equipotential level lines of a capacitor are quasicircles whose distortion depends only on the capacity and the level. As an application we find that given disjoint, nonseparating and nontrivial continua $E$ and $F$ in $\hat{\mathbb{C} }=\mathbb{C} \cup\{\infty\}$, the closed hyperbolic geodesic generating the fundamental group $\pi_1\big(\hat{\mathbb{C} }\setminus (E\cup F) \big) \cong \hat{\mathbb{Z} }$ is a $K$-quasicircle separating $E$ and $F$ with explicit distortion bound depending only on the capacity of $\hat{\mathbb{C} }\setminus (E\cup F)$. This result is then extended to obtain distortion bounds on a quasicircle representing a given homotopy class of a simple closed curve in a planar domain. Finally we are able to use these results to show that a simple closed hyperbolic geodesic in a planar domain is a quasicircle with a distortion bound depending explicitly, and only, on its length.

Estimates for Weierstrass division in ultradifferentiable classes

We study the Weierstrass division theorem for function germs in strongly non-quasianalytic Denjoy–Carleman classes CM. For suitable divisors P(x,t)=xd+a1(t)xd−1+⋯+ad(t) with real-analytic coefficients aj, we show that the quotient and the remainder can be chosen of class CMσ, where Mσ=((Mj)σ)j≥0 and σ is a certain Łojasiewicz exponent related to the geometry of the roots of P and verifying 1≤σ≤d. We provide various examples for which σ is optimal, in particular strictly less than d, which sharpens earlier results of Bronshtein and of Chaumat–Chollet.

The Euler obstruction of a function on a determinantal variety and on a curve

Given an analytic function germ f: (X, 0) → ℂ on an isolated determinantal singularity or on a reduced curve, we present formulas relating the local Euler obstruction of f to the vanishing Euler characteristic of the fiber X ∩ f-1(0) and to the Milnor number of f. Restricting ourselves to the case where X is a complete intersection, we obtain an easy way to calculate the local Euler obstruction of f as the difference between the dimension of two algebras.

Complex continuations of ℝan,exp-definable unary functions with a diophantine application

Let F denote the field of germs at of R an , exp -definable unary functions. Starting from its characterization in terms of closure conditions as given by van den Dries, Macintyre and Marker (‘The elementary theory of restricted analytic fields with exponentiation’, Ann. of Math. 140 (1994) 183–205; ‘Logarithmic-exponential power series’, J. London Math. Soc. (2) 56 (1997) 417–434), we give a similar description of its subring consisting of the germs of polynomial growth. More precisely, denoting this ring by F poly and its unique maximal ideal by m poly , our description picks out a subfield R poly of representatives of the residue field of F poly modulo m poly . In fact, such a construction, in considerably greater generality, was already carried out in 1997 by Kuhlmann and Kuhlmann (unpublished; see arXiv:1206.0711v1 [math.LO]) using valuation-theoretic methods, but our main aim here is to investigate the complex extensions of the functions under consideration. It turns out that R poly consists precisely of those (germs of) R an , exp -definable unary functions that have an R an , exp -definable analytic continuation to a right half-plane of C, and we use this fact to give a different proof of the Kuhlmann result. (Roughly speaking, the (real) valuation theory is replaced by the Phragmén–Lindelöf method applied to the complex continuations.) We then consider the analogous situation for those (germs of) functions in F having at most exponential growth. We briefly describe the Kuhlmann representation of the residue field and, although the functions therein do not have analytic continuations to a right half-plane in general, they turn out to have very good approximations that do. There has been much research over the last ten years on diophantine properties of sets definable in o-minimal structures (see, for example, L. A. Butler, ‘Some cases of Wilkie's conjecture’, Bull. London Math. Soc. 44 (2012) 642–660; G. O. Jones and M. E. M. Thomas, ‘The density of algebraic points on certain Pfaffian surfaces’, Q. J. Math. 63 (2012) 637–651; G. O. Jones, M. E. M. Thomas and A. J. Wilkie, ‘Integer-valued definable functions’, Bull. London Math. Soc. 44 (2012) 1285–1291; J. Pila and A. J. Wilkie, ‘The rational points of a definable set’, Duke Math. J. 133 (2006) 591–616; A. J. Wilkie, ‘Diophantine properties of sets definable in o-minimal structures’, J. Symbolic Logic 69 (2004) 851–861) and in the final section of this paper we make a small contribution to this work. We apply our results to adapt a method of Pólya (Über ganze ganzwertige Funktionen, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (1921) 1–10), as modified by Langley (J. K. Langley, ‘Integer-valued analytic functions in a half-plane’, Comput. Methods Funct. Theory 7 (2007) 433–442), and prove the following generalization of a result from G. O. Jones, M. E. M. Thomas and A. J. Wilkie, ‘Integer-valued definable functions’, Bull. London Math. Soc. 44 (2012) 1285–1291: let f : R → R be an R an , exp -definable function such that, for some r ∈ R with, | f ( x ) | ⩽ 2 r x for all sufficiently large x. Assume also that f ( n ) ∈ Z for all sufficiently large n ∈ N . Then there exists a polynomial P with rational coefficients such that f ( x ) = P ( x ) for all sufficiently large x.

A BOUND FOR THE MILNOR SUM OF PROJECTIVE PLANE CURVES IN TERMS OF GIT

Abstract. Let C be a projective plane curve of degree d whose singu-larities are all isolated. Suppose C is not concurrent lines. P loski provedthat the Milnor number of an isolated singlar point of C is less than orequal to (d−1) 2 −⌊ d2 ⌋. In this paper, we prove that the Milnor sum of Cis also less than or equal to (d− 1) 2 − ⌊ d2 ⌋ and the equality holds if andonly if C is a P loski curve. Furthermore, we find a bound for the Milnorsum of projective plane curves in terms of GIT. 1. IntroductionLet C = V (f) be a projective plane curve of degree d. In this paper, a planecurve C means a projective plane curve that has at most isolated singularities.Moreover, we assume that C is not concurrent lines. We assume that the basefield k is algebraically closed and char(k)=0. Let f = 0 at [0,0,1]. Then, wedefine its Milnor number at 0 (in the sense of affine chart) byµ 0 (f) = dim k (O 0 /J f ),where O 0 is a function germ of f at the origin and J f = (∂f/∂x,∂f/∂y) isthe Jacobian ideal of f. Since µ

Geometry and equisingularity of finitely determined map germs from $${\mathbb {C}}^n$$ C n to $${\mathbb {C}}^3$$ C 3 , $$n &gt;2$$ n &gt; 2

In this article we describe the geometry and the Whitney equisingularity of finitely determined map germs \(f{:} ({\mathbb {C}}^n,0) \rightarrow ( {\mathbb {C}}^3,0)\) with \(n \ge 3\). In the study of the geometry, we first investigate the critical locus \(\Sigma (f)\) of the germ, which is in the source. Then the discriminant \(\Delta (f)\), the image of the critical locus by the germ f, is studied. Last, but not least we investigate the set X(f), which is the inverse image by f of the discriminant. If the critical locus is not empty, the set X(f) is an hypersurface in the source that has nonisolated singularity at the origin. Concerning the Whitney equisingularity of families, we use some of the properties of the strata to prove that the Whitney equisingularity of an unfolding F is equivalent to the constancy of the Le numbers of the hypersurfaces \(\Delta (f)\) and X (f). From this study we describe some relationship among the invariants needed to describe the Whitney equisingularity of families in these dimensions, we reduce the number of invariants needed to a total of \(2n+2\), which improves substantially the number required by Gaffney’s theorem.

Moduli spaces for topologically quasi-homogeneous functions

We consider the topological class of a germ of 2-variables quasi-homogeneous complex analytic function. Each element f in this class induces a germ of foliation (f = constants) and a germ of curve (f = 0). We first describe the moduli space of the foliations in this class and give analytic normal forms. The classification of curves induces a distribution on this moduli space. By studying the infinitesimal generators of this distribution, we can compute the generic dimension of the moduli space for the curves, and we obtain the corresponding generic normal forms.

New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations

We use topology of configuration spaces to give a characterization of Neuwirth--Stallings pairs $(S^5, K)$ with $\dim K = 2$. As a consequence, we construct polynomial map germs $(\mathbb{R}^{6},0)\to (\mathbb{R}^{3},0)$ with an isolated singularity at the origin such that their Milnor fibers are not diffeomorphic to a disk, thus putting an end to Milnor's non-triviality question. Furthermore, for a polynomial map germ $(\mathbb{R}^{2n},0) \to (\mathbb{R}^{n},0)$ or $(\mathbb{R}^{2n+1},0) \to (\mathbb{R}^{n},0)$, $n \geq 3$, with an isolated singularity at the origin, we study the conditions under which the associated Milnor fiber has the homotopy type of a bouquet of spheres. We then construct, for every pair $(n, p)$ with $n/2 \geq p \geq 2$, a new example of a polynomial map germ $(\mathbb{R}^n,0) \to (\mathbb{R}^p,0)$ with an isolated singularity at the origin such that its Milnor fiber has the homotopy type of a bouquet of a positive number of spheres.

AKAP9, a Regulator of Microtubule Dynamics, Contributes to Blood-Testis Barrier Function

The blood-testis barrier (BTB), formed between adjacent Sertoli cells, undergoes extensive remodeling to facilitate the transport of preleptotene spermatocytes across the barrier from the basal to apical compartments of the seminiferous tubules for further development and maturation into spermatozoa. The actin cytoskeleton serves unique structural and supporting roles in this process, but little is known about the role of microtubules and their regulators during BTB restructuring. The large isoform of the cAMP-responsive scaffold protein AKAP9 regulates microtubule dynamics and nucleation at the Golgi. We found that conditional deletion of Akap9 in mice after the initial formation of the BTB at puberty leads to infertility. Akap9 deletion results in marked alterations in the organization of microtubules in Sertoli cells and a loss of barrier integrity despite a relatively intact, albeit more apically localized F-actin and BTB tight junctional proteins. These changes are accompanied by a loss of haploid spermatids due to impeded meiosis. The barrier, however, progressively reseals in older Akap9 null mice, which correlates with a reduction in germ cell apoptosis and a greater incidence of meiosis. However, spermiogenesis remains defective, suggesting additional roles for AKAP9 in this process. Together, our data suggest that AKAP9 and, by inference, the regulation of the microtubule network are critical for BTB function and subsequent germ cell development during spermatogenesis.