2,715 research outputs found

    A Subdivision Approach to Planar Semi-algebraic Sets

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    International audienceSemi-algebraic sets occur naturally when dealing with implicit models and boolean operations between them. In this work we present an algorithm to efficiently and in a certified way compute the connected components of semi-algebraic sets given by intersection or union of conjunctions of bi-variate equalities and inequalities. For any given precision, this algorithm can also provide a polygonal and isotopic approximation of the exact set. The idea is to localize the boundary curves by subdividing the space and then deduce their shape within small enough cells using only boundary information. Then a systematic traversal of the boundary curve graph yields polygonal regions isotopic to the connected components of the semi-algebraic set. Space subdivision is supported by a kd-tree structure and localization is done using Bernstein representation. We conclude by demonstrating our C++ implementation in the CAS Mathemagix

    Universality theorems for inscribed polytopes and Delaunay triangulations

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    We prove that every primary basic semialgebraic set is homotopy equivalent to the set of inscribed realizations (up to M\"obius transformation) of a polytope. If the semialgebraic set is moreover open, then, in addition, we prove that (up to homotopy) it is a retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of Q\mathbb{Q} are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mn\"ev universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, our results imply that the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.Comment: 15 pages, 2 figure

    Tropical approach to Nagata's conjecture in positive characteristic

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    Suppose that there exists a hypersurface with the Newton polytope Δ\Delta, which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of Δ\Delta to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of Δ\Delta from below. As a particular application of our method we consider a planar algebraic curve CC which passes through generic points p1,,pnp_1,\dots,p_n with prescribed multiplicities m1,,mnm_1,\dots,m_n. Suppose that the minimal lattice width ω(Δ)\omega(\Delta) of the Newton polygon Δ\Delta of the curve CC is at least max(mi)\max(m_i). Using tropical floor diagrams (a certain degeneration of p1,,pnp_1,\dots, p_n on a horizontal line) we prove that area(Δ)12i=1nmi2S,  where S=12max(i=1nsi2simi,i=1nsiω(Δ)).\mathrm{area}(\Delta)\geq \frac{1}{2}\sum_{i=1}^n m_i^2-S,\ \ \text{where } S=\frac{1}{2}\max \left(\sum_{i=1}^n s_i^2 \Big| s_i\leq m_i, \sum_{i=1}^n s_i\leq \omega(\Delta)\right). In the case m1=m2==mω(Δ)m_1=m_2=\ldots =m\leq \omega(\Delta) this estimate becomes area(Δ)12(nω(Δ)m)m2\mathrm{area}(\Delta)\geq \frac{1}{2}(n-\frac{\omega(\Delta)}{m})m^2. That rewrites as d(n1212n)md\geq (\sqrt{n}-\frac{1}{2}-\frac{1}{2\sqrt n})m for the curves of degree dd. We consider an arbitrary toric surface (i.e. arbitrary Δ\Delta) and our ground field is an infinite field of any characteristic, or a finite field large enough. The latter constraint arises because it is not {\it \`a priori} clear what is {\it a collection of generic points} in the case of a small finite field. We construct such collections for fields big enough, and that may be also interesting for the coding theory.Comment: major revision, many typos and mistakes are correcte

    Embeddings and immersions of tropical curves

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    We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we use our immersed tropical curves to construct totally faithful nodal algebraic curves via lifting results of Mikhalkin and Shustin.Comment: 23 pages, 14 figures, final submitted versio

    A bit of tropical geometry

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    This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking. Each definition is explained with concrete examples and illustrations. To a great exten, this text is an updated of a translation from a french text by the first author. There is also a newly added section highlighting new developments and perspectives on tropical geometry. In addition, the final section provides an extensive list of references on the subject.Comment: 27 pages, 19 figure

    Combinatorial models of expanding dynamical systems

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    We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally defined "Julia set" of the generalized dynamical systems depends only on the associated iterated monodromy group. We show then that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules.Comment: The new version differs substantially from the first one. Many parts are moved to other (mostly future) papers, the main open question of the first version is solve
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