Stepan Orevkov

Stepan Orevkov
Paul Sabatier University - Toulouse III | UPS Toulouse · Département de Mathématiques

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97
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Publications

Publications (97)
Preprint
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For a closed oriented surface $ \Sigma $ we define its degenerations into singular surfaces that are locally homeomorphic to wedges of disks. Let $X_{\Sigma,n}$ be the set of isomorphism classes of orientation preserving $n$-fold branched coverings $ \Sigma\rightarrow S^2 $ of the two-dimensional sphere. We complete $X_{\Sigma,n}$ with the isomorph...
Article
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We prove two inequalities for the complex orientations of a separating non-singular real algebraic curve in RP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb...
Conference Paper
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We are reproving the Matsuoka-Sakai inequality for plane curves, and give a global upper bound for the number of singular curves of a plane curve based on a local Zariski decomposition.
Article
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Funar algebra $K_\infty=K_\infty(\alpha,\beta;k)$ is the quotient of the group algebra over a ring $k$ of the braid group $B_\infty$ by two cubic relations: $\sigma_1^3-\alpha\sigma_1^2+\beta\sigma_1-1=0$ and another one which involves $\sigma_1$ and $\sigma_2$. The universal Markov trace on $K_\infty$ is the quotient map $t$ of $K_\infty(\alpha,\b...
Article
We define and calculate signature and nullity invariants for complex schemes for curves in the real projective plane. We use a generalization of the Murasugi-Tristram inequality to prohibit certain schemes from being realized by real algebraic curves.
Article
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We consider the Hurwitz action on quasipositive factorizations of a 3-braid. In a previous paper, for any given 3-braid we described a certain finite set which contains at least one representative of each orbit. Here we give an algorithm to decide if two elements of this finite set belong to the same orbit.
Article
Let $B'_n$ be the commutator subgroup of the braid group $B_n$. We prove that $Aut(B'_n)=Aut(B_n)$ for $n\ge5$ and $Aut(B'_4)$ is a semi-direct product of $Aut(B_4)$ and $Z/2Z$. This answers a question asked by Vladimir Lin.
Article
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We give an upper bound for the number of cusps of a plane affine or projective curve via its first Betti number.
Article
We give explicit parametric equations for all irreducible plane projective sextic curves which have at most double points and whose total Milnor number is maximal (is equal to 19). In each case we find a parametrization over a number field of the minimal possible degree and try to choose coordinates so that the coefficients are as small as we can d...
Article
We give an algorithm to decide if a given braid is a product of two factors which are conjugates of given powers of standard generators of the braid group. The same problem is solved in a certain class of Garside groups including Artin-Tits groups of spherical type. The solution is based on the Garside theory and, especially, on the theory of cycli...
Article
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We consider the Hurwitz action on quasipositive factorizations of 3-braids. We prove that every orbit contains an element of a special form. This fact provides an algorithm of finding representatives of every orbit for a given braid. We prove also that (1) any 3-braid has a finite number of orbits; (2) a Birman-Ko-Lee positive 3-braid has to most o...
Article
Full-text available
We give an algorithm to decide if a given braid is a product of two factors which are conjugates of given powers of standard generators of the braid group. The same problem is solved in a certain class of Garside groups including Artin-Tits groups of spherical type. The solution is based on the Garside theory and, especially, on the theory of cycli...
Article
Full-text available
Let $I_1\subset I_2\subset\dots$ be an increasing sequence of ideals of the ring $\Bbb Z[X]$, $X=(x_1,\dots,x_n)$ and let $I$ be their union. We propose an algorithm to compute the Gr\"obner base of $I$ under the assumption that the Gr\"obner bases of the ideal $\Bbb Q I$ of the ring $\Bbb Q[X]$ and the the ideals $I\otimes(\Bbb Z/m\Bbb Z)$ of the...
Article
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We want to describe the triplets (\Omega, (g), \mu) where (g) is the (co)metric associated to some symmetric second order differential operator L defined on the domain \Omega of R^d and such that L is expandable on a basis of orthogonal polynomials of L_2(\mu), and \mu is some admissible measure. Up to affine transformation, we find 11 compact doma...
Article
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The notion of basic net (called also basic polyhedron) on $S^2$ plays a central role in Conway's approach to enumeration of knots and links in $S^3$. Drobotukhina applied this approach for links in $\RP^3$ using basic nets on $\RP^2$. By a result of Nakamoto, all basic nets on $S^2$ can be obtained from a very explicit family of minimal basic nets...
Article
We propose a purely algebraic approach to construct invariants of transversal links in the standard contact structure on the 3-sphere generalizing Jones' approach to invariant of usual links. The only geometry used is the analogue of Alexander and Markov theorems. More precisely, we construct a trace on a certain cubic Hecke algebra which is invari...
Article
In this paper we construct several examples (series of examples) of real algebraic and real pseudoholomorphic curves in RP2 in which we tried to maximize different characteristics among curves of a given degree. In Sect. 2, this is the number of nonempty ovals; in Sect. 4, the number of ovals of the maximal depth; in Sect. 5, the number n such that...
Article
A study was conducted to describe commutator subgroups of artin groups. The study discussed when epimorphisms of commutator subgroups of Artin groups existed on nontrivial free groups. It was demonstrated that epimorphisms existed onto a non-trivial free group and an epimorphism of B '4 and D '4 onto a free group with two generators was obtained by...
Article
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For the finite groups GU(3), SU(3), GL(3), SL(3) over a finite field we solve the class product problem, i.e., we give a complete list of $m$-tuples of conjugacy classes whose product does not contain the identity matrix.
Article
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We prove complex orientation formulas for M-curves in RP2 of degree 4d + 1 with 4 nests. They generalize the formulas of complex orientations for M- curves in RP2 with a deep nest. This is a step towards the isotopy classification of real M-curves of degree 9.
Article
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The Agnihotri—Woodward—Belkale polytope Δ (resp., the Klyachko cone ) is the set of solutions of the multiplicative (resp., additive) Horn problem, i.e., the set of triples of spectra of special unitary (resp. traceless Hermitian) n × n matrices satisfying AB = C (resp. A + B = C). The set is the tangent cone of Δ at the origin. The group G = ℝ n...
Article
Full-text available
Under certain assumptions, the arrangements mentioned in the title are classified up to isotopy. Their algebraic realizability is discussed.
Article
π1(C2 - K) is computed, where K is an algebraic curve having only simple double points and satisfying certain restrictions at infinity. These restrictions are satisfied, for example, for a general curve parametrized by polynomials of given degrees, and also for a general curve with given Newton polyhedron. As a corollary, a new proof of the Fulton-...
Article
Let A be a real algebraic hypersurface in ℝ n+1 of degree d. For a point p∈A, we denote the curvature of A at p (that is, the Jacobian of the Gauss map γ:A→S n ) by k(p). Similarly, for a point p in the complexification ℂA of A, we denote the Gaussian curvature of ℂA at p by K(p). Using results of Teissier and Langevin, J.-J. Risler [Bull. Lond. Ma...
Article
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A negative answer is given to the following question of A.G. Vitushkin: Does there exist a nontrivial lower bound for the length of the maximal component of intersection of the unit sphere and an algebraic curve passing through the origin.
Article
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We prove that curves indicated in the title exist. This results answers to a question posed by A.G.Vitushkin about 30 years ago. We also discuss the minimal number of boundary components of a curve in the unit ball passing through the center, under the condition that all these components are shorter than a given number. More precisely, we discuss t...
Article
A braid is called quasipositive if it is a product of conjugates of standard generators of the braid group. We present an algorithm deciding if a given braid with three strings is quasipositive or not. The complexity (the time of work) of our algorithm is O(nk+1 )w heren is the length of the word in standard generators representing the braid and k...
Article
Singular braids are isotopy classes of smooth strings which are allowed to cross each other pairwise with distinct tangents. Under the usual multiplication of braids, they form a monoid. The singular braid group was introduced by R. Fenn, E. Keyman and C. Rourke [J. Knot Theory Ramifications 7, No. 7, 881-892 (1998; Zbl 0971.57011)] as the quotient...
Article
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We apply the Murasugi-Tristram inequality to real algebraic curves of odd degree in RP2 with a deep nest, i.e. a nest of the depth k -1 where 2k + 1 is the degree. For such curves, the ingredients of the Murasugi-Tristram inequality can be computed (or estimated) inductively using the computations for iterated torus links due to Eisenbud and Neuman...
Article
The number of topologically different plane real algebraic curves of a given degree $d$ has the form $\exp(C d^2 + o(d^2))$. We determine the best available upper bound for the constant $C$. This bound follows from Arnold inequalities on the number of empty ovals. To evaluate its rate we show its equivalence with the rate of growth of the number of...
Article
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The work presents some results on the asymptotics of the number of real plane algebraic curves as the degree grows. In particular, we obtain the asymptotics of the number of curves considered up to the isotopy and rigid isotopy, as well as the number of isotopic classes of maximal curves realizable by T-curves. Some results are generalized to hyper...
Article
On propose une methode de construction des courbes algebriques reelles planes donnees par y 3 +p(x)y+q(x) = 0, qui ont un arrangement prescrit sur le plan affine. La construction est basee sur la consideration de l'arrangement de f -1 (RP 1 ) sur CP 1 , ou f : CP 1 → CP 1 est le discriminant homogeneise, i.e. la fonction rationnelle definie par f(x...
Article
We show that there exists a real non-singular pseudoholomorphic sextic curve in the affine plane which is not isotopic to any real algebraic sextic curve. This result completes the isotopy classification of real algebraic affine M-curves of degree 6. Comparing this with the isotopy classification of real affine pseudoholomorphic sextic M-curves obt...
Article
We construct a plane real algebraic curve of degree 8 with 22 ovals (an M-curve) realizing the isotopy type $\left\langle {7 \sqcup 1\left\langle {2 \sqcup 1\left\langle {11} \right\rangle } \right\rangle } \right\rangle$ whose realizability was unknown.
Article
We prove that the union of a real algebraic curve of degree six and a real line on R P 2 \mathbf {RP}^{2} cannot be isotopic to the arrangement in Figure 1. Previously, the second author realized this arrangement with flexible curves. Here we show that these flexible curves are pseudo-holomorphic in a suitable tame almost complex structure on C P 2...
Article
We give an isotopy classification of real pseudo-holomorphic and real algebraic M-curves of degree 8 on the quadratic cone arranged in some special way with respect to a line, and show that there exist real pseudo-holomorphic curves which are not isotopic to any real algebraic curve in this class. In a similar way we find a pseudo-holomorphic real...
Article
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Let C⊂ℙ 2 be a rational curve of degree d which has only one analytic branch at each point. Denote by m the maximal multiplicity of singularities of C. It is proven by T. Matsuoka and F. Sakai [Math. Ann. 285, 233–247 (1989; Zbl 0661.14023)] that d<3m. We show that d<αm+const where α=2·61⋯ is the square of the “golden section”. We also construct ex...
Article
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It is shown that two braids represent transversally isotopic links if and only if one can pass from one braid to another by conjugations in braid groups, positive Markov moves, and their inverses.
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We construct a real surface of degree 5 in which has 23 connected components.RésuméOn construit une surface algébrique réelle de degré 5 dans qui a 23 composantes connexes.
Article
A quasipositive braid is any product of conjugates of the standard generators of the braid group, and a quasipositive link in S3 is isotopic to the closure of quasipositive braid. In this Note we prove that the boundary of an analytic curve in a pseudoconvex 4-ball is a quasipositive link. It was conjectured by Lee Rudolph.
Article
Dehornoy constructed a right invariant order on the braid group B n uniquely defined by the condition \(\beta _0 \sigma _i \beta _1 >1{\text{ if }}\beta _0 ,\beta _1\) are words in \(\sigma _{i + 1}^{ \pm 1} ,...,\sigma _{n - 1}^{ \pm 1}\). A braid is called strongly positive if \(\alpha \beta \alpha ^{ - 1} >1\) for any \(\alpha \in B_n\). In the...
Article
Let B-m = (sigma (1),...,sigma (m)|sigma (j)sigma (j+1)sigma (j) = sigma (j+1)sigma (j)sigma (j+1), [sigma (j),sigma (k)] = 1 for |k - j|>1) be the braid group. A braid b is called quasipositive if it has the form b = (a(1)sigma (1)a(1)(-1))...(a(k)sigma (1)a(k)(-1)). Using Gromov's theory of pseudo holomorphic curves, we prove that b epsilon B-m i...
Article
Let Bm = <σ1, . . . , σmσjσj+1σj = σj+1σjσj+1, [σj, σk] = 1 for k - j > 1> be the braid group. A braid b is called quasipositive if it has the form b= (a1σ1a1-1) ··· (akσ1ak-1). Using Gromov's theory of pseudo holomorphic curves, we prove that b ∈ Bm is quasipositive if and only if bσm ∈ Bm+1 is quasipositive. © 2000 Académie des sciences/Éditions...
Article
Full-text available
The nonsingular real plane algebraic curves of given degree $d$ are considered either up to isotopy or up to deformation. The asymptotic behavior of the number $I_d$ of isotopy classes and the number $D_d$ of deformation classes are studied. It is shown, in particular, that $log I_d\asypt d^2$. Other related problems and their higher dimensional ge...
Article
Let T2nbe the set of all triangulations of the square [0,n]2with all the vertices belonging toZ2. We show thatCn2
Article
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Let C ae IP 2 be an irreducible plane curve whose dual C ae IP 2 is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincar'e group G = ß 1 (IP 2 nC) contains a free group with two generators. If the geometric genus g of C is at least 2, this is clear because a subgroup of G can be mapped epimorphically...
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Without Abstract
Article
A braid is called algebraic if it is conjugated to the local braid of an algebraic function at a singular point. It is shown that any homomorphism of a free group into a braid group which takes each generator to an algebraic braid, can be realized as the braid monodromy of an algebraic function in a disk.
Article
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The paper is devoted to the Jacobian Conjecture: a polynomial mappingf22 with a constant nonzero Jacobian is polynomially invertible. The main result of the paper is as follows. There is no four-sheeted polynomial mapping whose Jacobian is a nonzero constant such that after the resolution of the indeterminacy points at infinity there is only one ad...
Article
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. The Harnack bound on the number of real components of a plane realalgebraic curve has a natural local version which states that the number of closedreal components obtained by a perturbation of a real isolated plane curve singularityhaving at least one real branch is bounded by the genus of the singularity (perturbationsattending this extremal va...
Article
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An example of a three-sheeted covering over the ball in 2 is constructed. The covering is the union of a four-dimensional ball with an analytic disk. The projection of the ball is a local homeomorphism ramified along the disk, whose projection is a part of an algebraic curve.
Article
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This is an extended, renovated and updated report on a joint work which the second named author presented at the Conference on Algebraic Geometry held at Saitama University, 15-17 of March, 1995. The main result is an inequality for the numerical type of singularities of a plane curve, which involves the degree of the curve, the multiplicities and...
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It is proved that the Jacobian of a 3-sheeted polynomial mapping cannot be a constant. Bibliography: 8 titles.
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