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We show that two bi-Lipschitz equivalent Brieskorn-Pham hypersurfaces have the same multiplicities at $0$. Moreover we show that if two algebraic $(n-1)$-dimensional cones $P, R\subset\mathbb C^n$ with isolated singularities are homeomorphic, then they have the same degree., Comment: 5 pages

In this article, we show the H\"older invariance of the Henry-Parusinski invariant. For a single germ $ f$, the Henry-Parusinski invariant of $ f $ is given in terms of the leading coefficients of the asymptotic expansion of $ f $ along the branches of the generic polar curve of $f$. As a consequence, we obtain that the classification problem of polynomial function-germs, with uniformly bounded degree, under H\"older equivalence, admits continuous moduli., Comment: 19 pages

We show that for every $k\ge 3$ there exist complex algebraic cones of dimension $k$ with isolated singularities, which are bi-Lipschitz and semi-algebraically equivalent but they have different degrees. We also prove that homeomorphic projective hypersurfaces with dimension greater than 2 have the same degree. In the final part of the paper, we classify links of real cones with base $\mathbb{P}^1\times \mathbb{P}^2.$ As an application we give an example of three four dimensional real algebraic cones in $\mathbb{R}^8$ with isolated singularity which are semi-algebraically and bi-Lipschitz equivalent but they have non-homeomorphic bases., Comment: 13 pages. arXiv admin note: text overlap with arXiv:2302.05387

In the paper \cite{renato} Renato Targino shows that bi-Lipschitz type of plane curve is determined by the local ambient topological properties of curves. Here we show that it is not longer true in higher dimensions. However we show that bi-Lipschitz type of space curves is determined by the number of singular points and by the local ambient topological type of a generic projection of such curves into the affine plane., Comment: 5 pages

The multiplicity of an algebraic curve $C$ in the complex plane at a point $p$ on that curve is defined as the number of points that occur at the intersection of $C$ with a general complex line that passes close to the point $p$. It is shown that $p$ is a singular point of the curve $C$ if and only if this multiplicity is greater than or equal to 2, in this sense, such an integer number can be considered as a measure of how singular can be a point of the curve $C$. In these notes, we address the classical concept of multiplicity of singular points of complex algebraic sets (not necessarily complex curves) and we approach the nature of the multiplicity of singular points as a geometric invariant from the perspective of the Multiplicity Conjecture (Zariski 1971). More precisely, we bring a discussion on the recent results obtained jointly with Lev Birbrair, Javier Fern\'andez de Bobadilla, L\^e Dung Trang and Mikhail Verbitsky on the bi-Lipschitz invariance of the multiplicity., Comment: 36 pages and 1 figure

We classify semi-algebraic surfaces in $\mathbb{R}^n$ with isolated singularities up to bi-Lipschitz homeomorphisms with respect to the inner distance. In particular, we obtain complete classifications for the Nash surfaces and the complex algebraic curves. We also address the minimal surfaces with finite total curvature., Comment: 20 pages and 2 figures

The paper is devoted to metric properties of singularities. We investigate the relations among topology, metric properties and smoothness. In particular, we present some higher dimensional analogous of Mumford's theorem on smoothness of normal surfaces. For example, we prove that a complex analytic set, with an isolated singularity at $0$, is smooth at $0$ if and only if it is locally metrically conical at $0$ and its link at $0$ is a homotopy sphere.

In this paper, we prove Fukui-Kurdyka-Paunescu's Conjecture, which says that subanalytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of the multiplicity (resp. degree) of real and complex analytic (resp. algebraic) sets. For instance, still in the real case, we prove a global version of Fukui-Kurdyka-Paunescu's Conjecture. In the complex case, one of the results that we prove is the following: If $(X,0)\subset (\mathbb{C}^n,0), (Y,0)\subset (\mathbb{C}^m,0)$ are germs of analytic sets and $h\colon (X,0)\to (Y,0)$ is a semi-bi-Lipschitz homeomorphism whose graph is a complex analytic set, then the germs $(X,0)$ and $(Y,0)$ have the same multiplicity. One of the results that we prove in the global case is the following: If $X\subset \mathbb{C}^n, Y\subset \mathbb{C}^m$ are algebraic sets and $\phi\colon X\to Y$ is a semialgebraic semi-bi-Lipschitz homeomorphism such that the closure of its graph in $\mathbb{P}^{n+m}(\mathbb{C})$ is an orientable homological cycle, then ${\rm deg}(X)={\rm deg}(Y)$., Comment: 19 pages. Some typos fixed and a small change in the title

We address the question of the bi-Lipschitz local triviality of a complex polynomial function over a complex value. Our main result state that a non constant complex polynomial admits a locally bi-Lipschitz trivial value if and only if it is a polynomial in a single complex variable.

It was conjectured that multiplicity of a singularity is bi-Lipschitz invariant. We disprove this conjecture, constructing examples of bi-Lipschitz equivalent complex algebraic singularities with different values of multiplicity., Comment: 8 pages

We study invariance of multiplicity of complex analytic germs and degree of complex affine sets under outer bi-Lipschitz transformations (outer bi-Lipschitz homeomorphims of germs in the first case and outer bi-Lipschitz homeomorphims at infinity in the second case). We prove that invariance of multiplicity in the local case is equivalent to invariance of degree in the global case. We prove invariance for curves and surfaces. In the way we prove invariance of the tangent cone and relative multiplicities at infinity under outer bi-Lipschitz homeomorphims at infinity, and that the abstract topology of a homogeneous surface germ determines its multiplicity., Comment: 11 pages

We prove that any complex analytic set in $\mathbb{C}^n$ which is Lipschitz normally embedded at infinity and has tangent cone at infinity that is a linear subspace of $\mathbb{C}^n$ must be an affine linear subspace of $\mathbb{C}^n$ itself. No restrictions on the singular set, dimension nor codimension are required. In particular, a complex algebraic set in $\mathbb{C}^n$ which is Lipschitz regular at infinity is an affine linear subspace., Comment: Revision in the statement of Corollary 5.7 and it is added Example 5.8. 12 pages

We prove that tangent cones of Lipschitz normally embedded sets are Lipschitz normally embedded. We also extend to real subanalytic sets the notion of reduced tangent cone and we show that subanalytic Lipschitz normally embedded sets have reduced tangent cones. In particular, we get that Lipschitz normally embedded complex analytic sets have reduced tangent cones., Comment: The paper was reorganized and also it was added an appendix by Anne Pichon and Walter D. Neumann. It has 13 pages and 6 figures

We prove that if two germs of irreducible complex analytic curves at $0\in\mathbb{C}^2$ have different sequence of characteristic exponents, then there exists $0<\alpha<1$ such that those germs are not $\alpha$-H\"older homeomorphic. For germs of complex analytic plane curves with several irreducible components we prove that if any two of them are $\alpha$-H\"older homeomorphic, for all $0<\alpha<1$, then there is a correspondence between their branches preserving sequence of characteristic exponents and intersection multiplicity of pair of branches. In particular, we recovery the sequence of characteristic exponents of the branches and intersection multiplicity of pair of branches are Lipschitz invariant of germs of complex analytic plane curves., Comment: 12 pages. Minor corrections in Example 4.16

Two blow-analytically equivalent real analytic plane function germs are sub-analytically bi-Lipschitz contact equivalent

We give partial answers to a metric version of Zariski's multiplicity conjecture. In particular, we prove the multiplicity of complex analytic surface (not necessarily isolated) singularities in $\mathbb{C}^3$ is a bi-Lipschitz invariant., Comment: Accepted for publication by the Journal of Topology

In this paper we study Lipschitz contact equivalence of continuous function germs in the plane definable in a polynomially bounded o-minimal structure, such as semialgebraic and subanalytic functions. We partition the germ of the plane at the origin into zones where the function has explicit asymptotic behavior. Such a partition is called a pizza. We show that each function germ admits a minimal pizza, unique up to combinatorial equivalence. We show then that two definable continuous function germs are definably Lipschitz contact equivalent if and only if their corresponding minimal pizzas are equivalent., Comment: 13 pages, 2 figures, added Section 5 on Geometric realization of abstract pizzas

The classical Theorem of Mumford states that a topologically regular complex algebraic surface in $\mathbb{C}^3$ with an isolated singular point is smooth. We proof that any Lipschitz regular complex algebraic set is smooth. No restriction on the dimension is needed. No restriction of singularity to be isolated is needed., Comment: This is a minor revision of the last version

In this short note, we consider the problem of bi-Lipschitz contact equivalence of complex analytic function-germs of two variables. It is inquiring about the infinitesimal sizes of such function-germs, up to bi-Lipschitz changes of coordinates. We show that this problem is equivalent to the problem of the right topological classification., Comment: The second version is substantially modified. 1) We have dropped section 5 of the first version. 2) The introduction is slightly modified. 3) Section 3 has been augmented with a correct statement of its main result (now referred as) Theorem 3.6. as a consequence of new Lemmas 3.1 and 3.2. Is also now given a proof of Proposition 3.7

We proof here the existence of a topological thick and thin decomposition of any closed definable thick isolated singularity germ in the spirit of the recently discovered metric thick and thin decomposition of complex normal surface singularities of [10]. Our thin zone catches exactly the homology of the family of the links collapsing faster than linearly. Simultaneously we introduce a class of rigid homeomorphisms more general than bi-Lipschitz ones, which map the topological thin zone onto the topological thin zone of its image. As a consequence of this point of view for the class of singularities we consider we exhibit an equivalent description of the notion of separating sets in terms of this fast contracting homology, Comment: 29 pp

We study the Lipschitz Geometry of Complex Algebraic Singularities. For this purpose we introduce the notion of choking horns. A Choking horn is a family of cycles on the family of the sections of an algebraic variety by very small spheres centered at a singular point, such that the cycles cannot be boundaries of nearby chains. The presence of choking horns is an obstruction to metric conicalness as we can see with some classical isolated hypersurfaces singularities which we prove are not metrically conic. We also show that there exist infinitely countably many singular varieties, which are locally homeomorphic, but not locally bi-Lipschitz equivalent with respect to the inner metric., Comment: 11 pages. Appendix by Walter D. Neumann

Let $Y:\R^n\to\R^n$ be a polynomial local diffeomorphism and let $S_Y$ denote the set of not proper points of $Y$. The Jelonek's real Jacobian Conjecture states that if $\codim(S_Y)\geq2$, then $Y$ is bijective. We prove a weak version of such conjecture establishing the sufficiency of a necessary condition for bijectivity. Furthermore, we generalize our result on bijectivity to semialgebraic local diffeomorphisms., Comment: 13 pages, 1 figure

The main goal of this work is to show that if two weighted homogeneous (but not homogeneous) function-germs $(\C^2,0)\to(\C,0)$ are bi-Lipschitz equivalent, in the sense that these function-germs can be included in a strongly bi-Lipschitz trivial family of weighted homogeneous function-germs, then they are analytically equivalent.

We show the existence of ($1+\frac{w_2}{w_3}$)-fast loops on semi-weighted homogeneous hypersurface singularities with weights $w_1\geq w_2>w_3$. In particular we show that semi-weighted homogeneous hypersurface singularities have metrical conical structure only if its two low weights are equal.

An explanation is given for the initially surprising ubiquity of separating sets in normal complex surface germs. It is shown that they are quite common in higher dimensions too. The relationship between separating sets and the geometry of the metric tangent cone of Bernig and Lytchak is described. Moreover, separating sets are used to show that the inner Lipschitz type need not be constant in a family of normal complex surface germs of constant topology., Comment: 12 pages. This paper includes the results of arXiv:0809.0845

We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense of semialgebraic or subanalytic sets) with image a complex algebraic surface., Comment: 5 pages. Submitted to Proceedings of the 10th International Workshop on Real and Complex Singularities

We show that a family of isolated complex hypersurface singularities with constant Milnor number may fail, in the strongest sense, to have constant bi-Lipschitz type. Our example is the Briac con--Speder family $X_t:=\{(x,y,z)\in\C^3 | x^5+z^{15}+y^7z+txy^6=0 \}$ of normal complex surface germs; we show the germ $(X_0, 0)$ is not bi-Lipschitz homeomorphic with respect to the inner metric to the germ $(X_t,0)$ for $t\ne 0$., Comment: 8 pages, 1 figure

We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface which particular emphasis on when it is metrically conical., Comment: Written for the proceedings of IV Iberoamerican Conference on Complex Geometry 2007

We compute Hoelder Complexes,i.e. the complete bi-Lipschitz invariants, for germs of real weighed homogeneous algebraic or semialgebraic surfaces., Comment: 11 pages

We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conic, i.e. the germs of the surfaces near singular points are not bi-Lipschitz equivalent, with respect to the inner metric, to cones. The technique used to prove the nonexistence of the metric conic structure is related to a development of Metric Homology. The class of the examples is rather large and it includes some surfaces of Brieskorn., Comment: 12 pages

We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz equivalent., Comment: 5 pages. Added result that nonhomogeneous cyclic quotients are not conical