Isotopic triangulation of a real algebraic surface (original) (raw)

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Let P ∈ Z[X, Y ] be a square-free polynomial and C(P) := {(α, β) ∈ R 2 , P (α, β) = 0} be the real algebraic curve defined by P. Our main result is an algorithm for the computation of the local topology in a neighbourhood of each of the singular points and critical points of the projection wrt the X-axis inÕ(d 6 τ +d 7) bit operations whereÕ means that we ignore logarithmic factors in d and τ. Compared to state of the art sub-algorithms used for computing a Cylindrical Algebraic Decomposition, this result avoids a generic shear and gives a deterministic algorithm for the computation of the topology of C(P) i.e a straight-line planar graph isotopic to C(P) inÕ(d 6 τ + d 7) bit operations.

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Oberwolfach Reports, 2008

Exercise: Show that any two geometric realizations are homeomorphic. (**) the link of every i-simplex triangulates a sphere of dimension d − i − 1. Caveat: Not every triangulation of a manifold satisfies condition (**). Exercise: If K satisfies condition (**) then so does Sd(K).

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We study the ideal triangulation graph T (S) of a punctured surface S of finite type. We show that if S is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of S into the simplicial automorphism group of T (S) is an isomorphism. We also show that under the same conditions on S, the graph T (S) equipped with its natural simplicial metric is not Gromov hyperbolic. Thus, from the point of view of Gromov hyperbolicity, the situation of T (S) is different from that of the curve complex of S.

Irrational connected sums and the topology of algebraic surfaces

Transactions of the American Mathematical Society, 1979

Suppose W is an irreducible nonsingular projective algebraic 3-fold and V a nonsingular hypersurface section of W. Denote by Vm a nonsingular element of \mV\. Let Vx, Vm, Vm + X be generic elements of \V\, \mV\, \{m + l)V\ respectively such that they have normal crossing in W. Let SXm = K, n Vm and C=K,nf"n Vm+l. Then SXm is a nonsingular curve of genus gm and C is a collection of N = m(m + \)VX points on SXm. By [MM2] we find that (•) Vm + i is diffeomorphic to Vm-r(5lm) U, V[-7"(S;m) where T(Slm) is a tubular neighborhood of 5lm in Vm, V[ is Vx blown up along C, S{" is the strict image of SXm in V{, T(S'Xm) is a tubular neighborhood of S[m in V'x and tj: dT(SXm)-»37\S¿) is a bundle diffeomorphism. Now V{ is well known to be diffeomorphic to Vx # N(-CP2) (the connected sum of Vx and N copies of CP2 with opposite orientation from the usual). Thus in order to be able to inductively reduce questions about the structure of Vm to ones about Vx we must simplify the "irrational sum" (•) above. The general question we can ask is then the following: Suppose A/, and M2 are compact smooth 4-manifolds and K is a connected ^-complex embedded in Aí¡. Let 7]; be a regular neighborhood of K in M¡ and let tj: 371,->3r2 be a diffeomorphism: Set V = M,-T, u M2-T2. How can the topology of V be described more simply in terms of those of M, and M2. In this paper we show how surgery can be used to simplify the structure of V in the case q = 1, 2 and indicate some applications to the topology of algebraic surfaces.

On the isotopic meshing of an algebraic implicit surface

Journal of Symbolic Computation, 2012

We present a new and complete algorithm for computing the topology of an algebraic surface S given by a squarefree polynomial in ◗[X,Y,Z]. Our algorithm involves only subresultant computations and entirely relies on rational manipulation, which makes it direct to implement. We extend the work in [15], on the topology of non-reduced algebraic space curves, and apply it to the polar curve or apparent contour of the surface S. We exploit simple algebraic criterion to certify the pseudo-genericity and genericity position of the surface. This gives us rational parametrizations of the components of the polar curve, which are used to lift the topology of the projection of the polar curve. We deduce the connection of the two-dimensional components above the cell defined by the projection of the polar curve. A complexity analysis of the algorithm is provided leading to a bound in e OB(d 15 τ) for the complexity of the computation of the topology of an implicit algebraic surface defined by integer coefficients polynomial of degree d and coefficients size τ. Examples illustrate the implementation in Mathemagix of this first complete code for certified topology of algebraic surfaces.

An Improved Upper Complexity Bound for the Topology Computation of a Real Algebraic Plane Curve

Journal of Complexity, 1996

The computation of the topological shape of a real algebraic plane curve is usually driven by the study of the behavior of the curve around its critical points (which includes also the singular points). In this paper we present a new algorithm computing the topological shape of a real algebraic plane curve whose complexity is better than the best algorithms known. This is due to the avoiding, through a sufficiently good change of coordinates, of real root computations on polynomials with coefficients in a simple real algebraic extension of ‫ޑ‬ to deal with the critical points of the considered curve. In fact, one of the main features of this algorithm is that its complexity is dominated by the characterization of the real roots of the discriminant of the polynomial defining the considered curve.

On the Topology of Real Algebraic Plane Curves

Mathematics in Computer Science, 2010

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on high-degree non-generic curves.

On the arc and curve complex of a surface

Mathematical Proceedings of the Cambridge Philosophical Society, 2010

We study the arc and curve complex AC(S) of an oriented connected surface S of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the simplicial automorphism group of AC(S) coincides with the natural image of the extended mapping class group of S in that group. We also show that for any vertex of AC(S), the combinatorial structure of the link of that vertex characterizes the type of a curve or of an arc in S that represents that vertex. We also give a proof of the fact if S is not a sphere with at most three punctures, then the natural embedding of the curve complex of S in AC(S) is a quasiisometry. The last result, at least under some slightly more restrictive conditions on S, was already known. As a corollary, AC(S) is Gromovhyperbolic.

On the computation of the topology of a non-reduced implicit space curve

Proceedings of the twenty-first international symposium on Symbolic and algebraic computation - ISSAC '08, 2008

An algorithm is presented for the computation of the topology of a non-reduced space curve defined as the intersection of two implicit algebraic surfaces. It computes a Piecewise Linear Structure (PLS) isotopic to the original space curve.

On the topology of planar algebraic curves

2009

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position.