In this paper we study Lipschitz contact equivalence of continuous function germs in the plane de... more 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.
In the paper we construct some stratifications of the space of monic polynomials in real and comp... more In the paper we construct some stratifications of the space of monic polynomials in real and complex cases. These stratifications depend on properties of roots of the polynomials on some given semialgebraic subset of R or C. We prove differential triviality of these stratifications. In the real case the proof is based on properties of the action of the group of interval exchange transformations on the set of all monic polynomials of some given degree. Finally we compare stratifications corresponding to different semialgebraic subsets.
Proceedings of the American Mathematical Society, 2015
The classical Theorem of Mumford states that a topologically regular complex algebraic surface in... more The classical Theorem of Mumford states that a topologically regular complex algebraic surface in 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.
Electronic Research Announcements in Mathematical Sciences, 2014
We prove that globally subanalytic nonsingular CMC surfaces of R 3 are only planes, round spheres... more We prove that globally subanalytic nonsingular CMC surfaces of R 3 are only planes, round spheres or right circular cylinders.
Page 1. arXiv:1105.3327v1 [math.AG] 17 May 2011 THE THICK-THIN DECOMPOSITION OF A NORMAL SURFACE ... more Page 1. arXiv:1105.3327v1 [math.AG] 17 May 2011 THE THICK-THIN DECOMPOSITION OF A NORMAL SURFACE SINGULARITY LEV BIRBRAIR, WALTER D NEUMANN, AND ANNE PICHON Abstract. We describe a natural ...
We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense... more 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.
We show that a generic semi-quasihomogeneous surface germ in R 3 with isolated singularity is bi-... more We show that a generic semi-quasihomogeneous surface germ in R 3 with isolated singularity is bi-Lipschitz homeomorphic, with respect to the inner metric, to its quasihomogeneous approximation.
The problem of semialgebraic Lipschitz classification of quasihomogeneous polynomials on a Hölder... more The problem of semialgebraic Lipschitz classification of quasihomogeneous polynomials on a Hölder triangle is studied. For this problem, the "moduli" are described completely in certain combinatorial terms.
Abstract Let $ P^{k}(n, 2) $ be the set of all real polynomial map germs $ f=(f_1, f_2):(\ mathbb... more Abstract Let $ P^{k}(n, 2) $ be the set of all real polynomial map germs $ f=(f_1, f_2):(\ mathbb R^ n, 0)\ rightarrow (\ mathbb R^ 2, 0) $ with degree of $ f_1, f_2\ leq k $. The main result of this paper shows that the set of equivalence classes of $ P^{k}(n, 2) $, with ...
... Lev Birbrair · Alexandre Fernandes Received ... Lev BirbrairResearch supported under CNPq Gr... more ... Lev Birbrair · Alexandre Fernandes Received ... Lev BirbrairResearch supported under CNPq Grant N 300985/93-2. Alexandre FernandesResearch supported under CNPq Grant N 300393/2005-9 and FUNCAP/CNPq/PPP. L ...
Communications on Pure and Applied Mathematics, 2008
We produce examples of complex algebraic surfaces with isolated singularities such that these sin... more 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 inner metric, to cones. The technique used to prove the nonexistence of the metric conic structure is related to a development of Metric Homology.
In this paper we show that the tangent cone of a conflict set in R n is a linear affine cone over... more In this paper we show that the tangent cone of a conflict set in R n is a linear affine cone over a conflict set of smaller dimension and has dimension n − 1. Moreover we give an example where the conflict sets is not normally embedded and not locally bi-Lipschitz equivalent to the corresponding tangent cone.
We compute Hoelder Complexes,i.e. the complete bi-Lipschitz invariants, for germs of real weighed... more We compute Hoelder Complexes,i.e. the complete bi-Lipschitz invariants, for germs of real weighed homogeneous algebraic or semialgebraic surfaces. Date: February 2, 2008.
An extremely important corollary of the Triangulation Theorem of Lojasiewicz [9] is the existen... more An extremely important corollary of the Triangulation Theorem of Lojasiewicz [9] is the existence of topologically conic structure near a singular point of an algebraic set (real or complex). Namely, Lo-jasiewicz proved the following: let X be an algebraic (or semialgebraic) set in ...
In this paper we study Lipschitz contact equivalence of continuous function germs in the plane de... more 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.
In the paper we construct some stratifications of the space of monic polynomials in real and comp... more In the paper we construct some stratifications of the space of monic polynomials in real and complex cases. These stratifications depend on properties of roots of the polynomials on some given semialgebraic subset of R or C. We prove differential triviality of these stratifications. In the real case the proof is based on properties of the action of the group of interval exchange transformations on the set of all monic polynomials of some given degree. Finally we compare stratifications corresponding to different semialgebraic subsets.
Proceedings of the American Mathematical Society, 2015
The classical Theorem of Mumford states that a topologically regular complex algebraic surface in... more The classical Theorem of Mumford states that a topologically regular complex algebraic surface in 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.
Electronic Research Announcements in Mathematical Sciences, 2014
We prove that globally subanalytic nonsingular CMC surfaces of R 3 are only planes, round spheres... more We prove that globally subanalytic nonsingular CMC surfaces of R 3 are only planes, round spheres or right circular cylinders.
Page 1. arXiv:1105.3327v1 [math.AG] 17 May 2011 THE THICK-THIN DECOMPOSITION OF A NORMAL SURFACE ... more Page 1. arXiv:1105.3327v1 [math.AG] 17 May 2011 THE THICK-THIN DECOMPOSITION OF A NORMAL SURFACE SINGULARITY LEV BIRBRAIR, WALTER D NEUMANN, AND ANNE PICHON Abstract. We describe a natural ...
We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense... more 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.
We show that a generic semi-quasihomogeneous surface germ in R 3 with isolated singularity is bi-... more We show that a generic semi-quasihomogeneous surface germ in R 3 with isolated singularity is bi-Lipschitz homeomorphic, with respect to the inner metric, to its quasihomogeneous approximation.
The problem of semialgebraic Lipschitz classification of quasihomogeneous polynomials on a Hölder... more The problem of semialgebraic Lipschitz classification of quasihomogeneous polynomials on a Hölder triangle is studied. For this problem, the "moduli" are described completely in certain combinatorial terms.
Abstract Let $ P^{k}(n, 2) $ be the set of all real polynomial map germs $ f=(f_1, f_2):(\ mathbb... more Abstract Let $ P^{k}(n, 2) $ be the set of all real polynomial map germs $ f=(f_1, f_2):(\ mathbb R^ n, 0)\ rightarrow (\ mathbb R^ 2, 0) $ with degree of $ f_1, f_2\ leq k $. The main result of this paper shows that the set of equivalence classes of $ P^{k}(n, 2) $, with ...
... Lev Birbrair · Alexandre Fernandes Received ... Lev BirbrairResearch supported under CNPq Gr... more ... Lev Birbrair · Alexandre Fernandes Received ... Lev BirbrairResearch supported under CNPq Grant N 300985/93-2. Alexandre FernandesResearch supported under CNPq Grant N 300393/2005-9 and FUNCAP/CNPq/PPP. L ...
Communications on Pure and Applied Mathematics, 2008
We produce examples of complex algebraic surfaces with isolated singularities such that these sin... more 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 inner metric, to cones. The technique used to prove the nonexistence of the metric conic structure is related to a development of Metric Homology.
In this paper we show that the tangent cone of a conflict set in R n is a linear affine cone over... more In this paper we show that the tangent cone of a conflict set in R n is a linear affine cone over a conflict set of smaller dimension and has dimension n − 1. Moreover we give an example where the conflict sets is not normally embedded and not locally bi-Lipschitz equivalent to the corresponding tangent cone.
We compute Hoelder Complexes,i.e. the complete bi-Lipschitz invariants, for germs of real weighed... more We compute Hoelder Complexes,i.e. the complete bi-Lipschitz invariants, for germs of real weighed homogeneous algebraic or semialgebraic surfaces. Date: February 2, 2008.
An extremely important corollary of the Triangulation Theorem of Lojasiewicz [9] is the existen... more An extremely important corollary of the Triangulation Theorem of Lojasiewicz [9] is the existence of topologically conic structure near a singular point of an algebraic set (real or complex). Namely, Lo-jasiewicz proved the following: let X be an algebraic (or semialgebraic) set in ...
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