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. [16]), 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**

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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 [3], P Cassou-Noguès, C Eyral and M Oka [8], A Degtyarev [9], M Oka [14], I Shimada [15], or the first author [11]. E Artal suggests in [1] 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.

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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:

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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

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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.

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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

(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.

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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
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

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109 En savoir plus

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 +

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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.

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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 [40] 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 [16], 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:

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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

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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

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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.

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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.

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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.

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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

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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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