# Topology of algebraic curves

## Top PDF Topology of algebraic curves: ### A subdivision arrangement algorithm for semi-algebraic curves: an overview

In the case of an implicit curve, we test that the Bernstein coeﬃcients of f k have no sign change to en- sure (1). For (2), we test that these coeﬃcients have at most one sign change on each side of D, and that the coeﬃcients of ∂ x f (resp. ∂ y f) have no sign change. For (3), we test that the coeﬃcients of f, ∂xf, ∂y f both change their sign and that the size of the box is small enough ( < ). This step for which we can use a subdivision solver which isolates the singular points of f(x, y) = 0 (e.g. ), allows us to deal safely with “approximate singular points”. These tests ensure us that the topology of a curve ok inside D is uniquely determined from the points of o k on the boundary of
En savoir plus ### A linking invariant for algebraic curves

1. Introduction The topological study of algebraic plane curves was initiated at the beginning of the 20th century by F Klein and H Poincaré. One of the main questions is to understand the relationship between the combinatorics and the topology of a curve. It is known, since the seminal work of O Zariski [18, 19, 20], that the topological type of the embedding of an algebraic curve in the complex projective plane is not determined by the combinatorics. Indeed, Zariski constructed two sextics with 6 cusps having same combinatorics, and proved that the fundamental group of their complements are not isomorphic. Geometrically, these two curves are distinguished by the fact that the cusps in the first curve lie on a conic, while they do not in the second one. Since this historical example, using various methods, numerous examples of pairs of algebraic curves having same combinatorics but different topology have been found, see for example E Artal, J I Cogol- ludo and H Tokunaga , P Cassou-Noguès, C Eyral and M Oka , A Degtyarev , M Oka , I Shimada , or the first author . E Artal suggests in  to call such examples Zariski pairs. The topology of curves in CP 2 is intimately connected to the topology of knots and links in S 3 . Several tools are indeed shared by these two domains, such as the homology or the fundamental group of the complement, the Alexander polynomial or module, etc, although they usually have rather different behaviors.
En savoir plus ### Regularity criteria for the topology of algebraic curves and surfaces

smooth on each stratum, and Π R m ◦ h = g. Now that we have acquired a proper understanding of how an algebraic va- riety can be decomposed into smooth strata that fit nicely together, we can actually take advantage of this decomposition to determine conditions sufficient to know the topology in small enough boxes. Indeed, as our algorithm proceeds by subdivision, we will finally end up with boxes as small as we want. So we just have to take care of what happens locally. We know we can split the surface into patches of 2-dimensional strata (the smooth part), 1-dimensional strata (singular curves), and 0-dimensional strata (singular points were the 1-dimensional strata do not meet condition B). Topologically we can characterize the topological sit- uation as follows:
En savoir plus ### Almost algebraic actions of algebraic groups and applications to algebraic representations

Our basic class of Polish actions will be given by actions of algebraic groups on algebraic varieties. As mentioned in Setups 1.1 & 1.3 , we fixed a complete and separable valued field .k; j  j/ , that is a field k with an absolute value j  j which is complete and separable (in the sense of having a countable dense subset). See [ 9 , 6 ] 1 for a general discussion on these fields. It is a standard fact that a complete absolute value on a field F has a unique extension to its algebraic closure F x ### The Complex Representation of Algebraic Curves and its Simple Exploitation for Pose Estimation and Invariant Recognition

Â¹ÆSÅÈ¿Á» É6ÞKÊ |Â ÆSÅÈ¿n» É6Þ Ñ ÊpÉYËÓØPÅÈ¿n¿nÅÝÂ|ÐfÉ`ÄÃnÄ Ñ ÉYË9ØPÅÈ¿}¿nÅÈÂ|ÐÉ¹ÄÃnÄ Ì Í ÏÞ ÎÐË ÎÑpÞKÊË pÑ Þ>ÑUË ÒUÎÞ>ÓUË ÃÁ½}ÄwÂ`¿nÏÈÄÃnÅÇÆSÂ Í >Þ ÑUË ÊÒpÞhÔË ÒpÞkÉYË ÕpÞkÉYË... phi[r] ### Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus

CONTAINING NO ABSOLUTELY IRREDUCIBLE CURVES OF LOW GENUS YVES AUBRY, ELENA BERARDINI, FABIEN HERBAUT AND MARC PERRET Abstract. We provide a theoretical study of Algebraic Geometry codes con- structed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that the abelian surface does not contain low genus curves. This approach naturally leads us to consider Weil restric- tions of elliptic curves and abelian surfaces which do not admit a principal polarization.
En savoir plus ### On Robust Fitting of Curves and Sets of Curves

The road marking model must be as simple as possible, yet it must also put up with the variability of markings aspect in real-world scenes. We propose several functional bases that are suitable for this task. The model itself is linear with respect to its parameters, which contributes to the simplicity of the approach. This linear, deterministic generative ### On the topology of planar algebraic curves

(the x-coordinates of these points are exactly the positions of a vertical sweep line at which there may be a change in the number of intersection points with C). It is thus useful to require that all the x-critical points of C are mapped to vertices of the graph we want to compute. To our knowledge, almost all methods for computing the topology of a curve compute the critical points of the curve and associate corresponding vertices in the graph. (Refer to [1, 9] for recent subdivision methods that avoid the compu- tation of non-singular critical points.) However, it should be stressed that almost all methods do not necessarily compute the critical points for the specified x-direction. Indeed, when the curve is not in generic position, that is, if two x-critical points have the same x-coordinate or if the curve admits a vertical asymptote, most algorithms shear the curve so that the resulting curve is in generic position. This is, however, an issue for several reasons. First, determining whether a curve is in generic position is not a trivial task and it is time consuming [23, 39]. Second, if one wants to compute ar- rangements of algebraic curves with a sweep-line approach, the extreme points of all the curves have to be computed for the same direction. Finally, if the coordinate system is sheared, the polynomial of the initial curve is transformed into a dense polynomial, which slows down, in practice, the computation of the critical points.
En savoir plus ### An explicit algebraic family of genus-one curves violating the Hasse principle

any of the 27 lines on V. (With more work, one could probably prove the general case too, but we have not tried too seriously, since the special case proved is all we need for our application, and also since the second proof works generally.) The second proof can be interpreted as explaining the ### Matrix-based implicit representations of algebraic curves and surfaces and applications

Matrix-based implicit representations of algebraic curves and surfaces and applications Abstract. In this thesis, we introduce and study a new implicit representation of rational curves of arbitrary dimensions and propose an implicit representation of rational hypersur- faces. Then, we illustrate the advantages of this matrix representation by addressing sev- eral important problems of Computer Aided Geometric Design (CAGD): The curve/curve, curve/surface and surface/surface intersection problems, the point-on-curve and inversion problems, the computation of singularities of rational curves. We also develop some sym- bolic/numeric algorithms to manipulate these new representations for example: the algorithm for extracting the regular part of a non square pencil of univariate polynomial matrices and bivariate polynomial matrices. In the appendix of this thesis work we present an implemen- tation of these methods in the computer algebra systems Mathemagix and Maple. In the last chapter, we describe an algorithm which, given a set of univariate polynomials f 1 , ..., f s
En savoir plus ### Tropical intersection theory, and real inflection points of real algebraic curves

These examples are constructed using Viro’s Patchworking method in the dense torus of the normal toric surface CP(2, 1, 1). In particular, we thoroughly study all possible distribution of real inflection points of real algebraic curves in (C ∗ ) 2 having some particular “small” Newton polygon. A generic algebraic curve C with Newton polygon the parallelogram with vertices (0, 2), (0, 1), (2, 1), and (2, 0) is non-singular of genus one. The tautological embedding of the normal toric surface CP(2, 1, 1) ,→ CP 3 defines a complete linear series Q of degree 4 and rank 3 on C by restricting O CP 3 (1). Thus Q has 16 inflection points with at most 8 of them real. The above construction relies on the following proposition. Proposition 0.3.6. The complete linear series Q of degree 4 and rank 3 on a non-singular real algebraic curve in CP(2, 1, 1) defined by a polynomial f (X, Y ) = u 02 Y 2 + (u 01 + u 11 X + u 21 X 2 )Y +
En savoir plus ### A certified numerical algorithm for the topology of resultant and discriminant curves

Up to homeomorphisms, the local topology at a point of a curve is characterized by the number of real (half-) branches of the curve connected to the point. This number is two for a regular point, and can be any even number at a singular point. We define a witness box of a singular point as a box containing the singular point such that the topology of the curve inside the box is the one of the graph connecting the singularity to the crossings of the curve with the box boundary. The topology of the curve in a witness box is thus completely determined by its number of crossings with the box boundary. In this article, we only focus on the first two steps of the above-mentioned topology algorithm, that is computing witness boxes. The third step can be seen as reporting the topology of a smooth curve in the complement of the witness boxes of the singular points. There already exist certified algorithms for this task using subdivision ( Plantinga and Vegter ( 2004 ); Lin and Yap ( 2011 )). Another option is to use certified path tracking (e.g. Martin et al. ( 2013 ); Van Der Hoeven ( 2011 ); Beltr´an and Leykin ( 2013 )) starting at the crossings of the curve with the boundary of singularity boxes, note that to report the closed loops without singularity, at least one point on such components must be provided. Contribution and overview.. The specificity of the resultant or the discriminant curves computed from generic surfaces is that their singularities are stable, this is a classical result of singularity theory due to Whitney. The key idea of our work is to show that, in this specific case, the over-determined system defining the curve singularities can be transformed into a regular well- constrained system of a transverse intersection of two curves defined by subresultants. This new formulation can be seen as a specific deflation system that does not contain spurious solutions.
En savoir plus ### Logarithms of algebraic numbers

In fact, we do not strive for new numerical improvements but want to expose the efficacy of ‘G-function techniques’ in transcendence theory. In this spirit, we use Siegel’s lemma for the construction of Padé approximants and hence have little information on them. In particular, we have no explicit formulas at our disposal. Therefore, it is not reasonable to expect the results to keep up numerically with the above results or even with more recent advances (e.g. those in [1, 30, 31]) on certain irrationality and transcendence measures of logarithms. However, the abstract approach here is a proper contribution to the scarcely developed transcendence theory of G-functions. The mathematical interest in this class of functions goes back to Siegel’s landmark article  on diophantine approximation. 1 In its first part about transcendental numbers he studied in detail the transcendence of values attained by E-functions at algebraic points. In this context, one finds the first definition of G-functions and some results on the diophantine nature of their values at algebraic points, mostly in terms of irrationality measures. However, Siegel did not present a proof of his announcements and it took over fifty years until results of this sort were proven in literature by Bombieri , triggering further research in this direction by various authors [2, 4, 20, 21]. But already Siegel (p. 240, ibid.) noticed a major deficiency of his announcements on G-functions with respect to transcendence:
En savoir plus ### Around rationality of algebraic cycles

where we write Y := Y F , with F an algebraic closure of F . We recall that an element y of Ch(Y ) is F (Q)-rational if its image y F (Q) under Ch(Y ) → Ch(Y F (Q) ) is in the image of Ch(Y F (Q) ) → Ch(Y F (Q) ). Since F is algebraically closed, the bottom homomorphism Ch(Y ) → Ch(Y F (Q) ) is injective by the specialization arguments. Furthermore, for any I ⊂ {0, . . . , [n/2]}, let us denote the associated partial flag variety as G(I) (in particular, for any i ∈ {0, . . . , [n/2]}, the variety G(i) is the Grassmannian of i-dimensional totally isotropic subspaces) and for J ⊂ I we write π with subindex I with J underlined inside it for the natural projection G(I) → G(J ). In particular, for any i ∈ {0, . . . , [n/2]}, one can consider
En savoir plus ### Constructions of real algebraic surfaces

sume that [Z] = (1, a, b), with a, b ∈ Z/2Z. Consider a real algebraic surface S of tridegree (1, a, b) transverse to Z. Then, one has [Z ∪S] = [Z]+[S] = 0, and the union Z ∪S bounds in (RP 1 ) 3 . Thus, one can color the complement (RP 1 ) 3 \ (Z ∪ S) into two colors in such a way that the components adjacent from the different sides to the same (two-dimensional) piece of Z ∪ S would be of different colors. It is a kind of checkerboard coloring. Consider the disjoint sum Q of the closures of those components of (RP 1 ) 3 \(Z ∪S) which are colored
En savoir plus ### Algebraic Analysis of Branching Processes

Finite prefixes of net unfoldings constitute a first trans- formation of the initial Petri Net (PN), where cycles have been flattened. This computation produces a process set where conflicts act as a discriminating factor. A conflict partitions a process in branching processes. An unfolding can be trans- formed into a set of finite branching processes. Theses pro- cesses constitute a set of acyclic graphs - several graphs can be produced when the PN contains parallelism - built with events and conditions, and structured with two operators: causality and true parallelism. An interesting particularity of an unfolding is that in spite of the loss of the concept of global marking, these processes contain enough information to reconstitute the reachable markings of the original Petri nets. In most of the cases, unfoldings are larger than the original Petri net. This is provoked essentially when values of precondition places exceed the precondition of non simple conflicts. This produces a lot of alternative conditions. In spite of that, a step has been taken forward: cycles have been broken and the conflicts have structured the nets in branching processes.
En savoir plus ### COMPLEXITY OF SEMI-ALGEBRAIC PROOFS

F c denotes constant-depth Frege systems. See Fig. 1 for other no- tation. Only the strongest separations relevant to semi-algebraic systems are shown. The leftmost separation is due to PHP (the positive part is proved in [Pud99], the negative part is proved in [Hak85]). The counterexample for CP (which provides the two sep- arations in the middle) is given by the clique-coloring tautologies (resp., Theorem 4.1 and [Pud97]). The two rightmost separations are due to Tseitin's formulas(resp., Theorem 6.1 and [BS02]). Note that the knapsack problem is not a valid counterexample because it is not a translation of a formula in DNF.
En savoir plus ### Algebraic Analysis of Branching Processes

Finite prefixes of net unfoldings constitute a first trans- formation of the initial Petri Net (PN), where cycles have been flattened. This computation produces a process set where conflicts act as a discriminating factor. A conflict partitions a process in branching processes. An unfolding can be trans- formed into a set of finite branching processes. Theses pro- cesses constitute a set of acyclic graphs - several graphs can be produced when the PN contains parallelism - built with events and conditions, and structured with two operators: causality and true parallelism. An interesting particularity of an unfolding is that in spite of the loss of the concept of global marking, these processes contain enough information to reconstitute the reachable markings of the original Petri nets. In most of the cases, unfoldings are larger than the original Petri net. This is provoked essentially when values of precondition places exceed the precondition of non simple conflicts. This produces a lot of alternative conditions. In spite of that, a step has been taken forward: cycles have been broken and the conflicts have structured the nets in branching processes.
En savoir plus ### Decomposition of s ― Concentration Curves

jB (p) − CC iB s−1 (p) (15) + αCC jT s−1 (p) − CC iT s−1 (p) ≥ 0, for all p ∈ [0, 1] . Proof. See the Appendix. Note that the specification of within-group contribution curves brings out the average within-group inequalities. It turns out that, it would be appeal- ing to formalize a taxation technique ensuring decision makers that welfare- improving tax reforms reduce inequalities within all subgroups. Indeed, this condition is not guaranteed in Theorem 4.1, for which within-group inequal- ities in mean may only be reduced for good j (if αCC jW s−1 dominates CC iW s−1 for α ≤ 1). Subsequently, if we were able to construct within-group contri- bution curves for all groups Π k , k ∈ {1, 2, . . . , K}, (say CC jW,k s−1 for the j-th
En savoir plus ### On the total order of reducibility of a pencil of algebraic plane curves

The first main result of this paper is that the upper bound d 2 − 1 for ρ(f, g) is also valid for m(f, g). This is the content of Section 2 where it is assumed that the characteristic of K is zero. Our method, which is inspired by [Rup86], is elementary compared to the previously mentioned papers. Roughly speaking, we will transform the pencil of curves into a pencil of matrices and obtain in this way the claimed bound as a consequence of rank computations of some matrices that we will study in Section 1. In this way, the known inequality ρ(f, g) ≤ d 2 − 1 is easily obtained. Moreover, we will
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