2,715 research outputs found
A Subdivision Approach to Planar Semi-algebraic Sets
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
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 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
Suppose that there exists a hypersurface with the Newton polytope ,
which passes through a given set of subvarieties. Using tropical geometry, we
associate a subset of to each of these subvarieties. We prove that a
weighted sum of the volumes of these subsets estimates the volume of
from below.
As a particular application of our method we consider a planar algebraic
curve which passes through generic points with prescribed
multiplicities . Suppose that the minimal lattice width
of the Newton polygon of the curve is at least
. Using tropical floor diagrams (a certain degeneration of
on a horizontal line) we prove that
In the case this estimate becomes
. That
rewrites as for the curves
of degree .
We consider an arbitrary toric surface (i.e. arbitrary ) 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
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
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
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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