Plane Algebraic Curves Research Articles

Exact symbolic–numeric computation of planar algebraic curves

We present a certified and complete algorithm to compute arrangements of real planar algebraic curves. It computes the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to compute the topology of a single algebraic curve.Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of symbolic operations involved, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry.We have implemented our algorithms as prototypical contributions to the C++-project Cgal. We exploit graphics hardware to expedite the remaining symbolic computations. We have also compared our implementation with the current reference implementations, that is, Lgp and Maple’s Isolate for polynomial system solving, and Cgal’s bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones.

Genus one factors of curves defined by separated variable polynomials

We give some sufficient conditions on complex polynomials P and Q to assure that the algebraic plane curve P(x)-Q(y)=0 has no irreducible component of genus 0 or 1. Moreover, if deg (P)=deg (Q) and if both P, Q satisfy Hypothesis I introduced by H. Fujimoto, our sufficient conditions are necessary.

A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements

Let C1 and C2 be algebraic plane curves in ℂ 2 such that the curves intersect in d1 · d2 points where d1, d2 are the degrees of the curves respectively. Oka and Sakamoto proved that π1( ℂ 2 \ C1 U C2)) ≅ π1 ( ℂ 2 \ C1) × π1 ( ℂ 2 \ C2) [1]. In this paper we prove the converse of Oka and Sakamoto’s result for line arrangements. Let A1 and A2 be non-empty arrangements of lines in ℂ 2 such that π1 (M(A1 U A2)) ≅ π1 (M(A1)) × π1 (M(A2)) Then, the intersection of A1 and A2 consists of /A1/ · /A2/ points of multiplicity two.

Laws of rolling simplexes (feild equations according to Tycho Brahe)

Models of central power fields with quadratic dynamics suggested by Rene Descartes are studied. These examples gradually lead the reader to the understanding of base notions of differen-tial algebraic-geometric theory of Brahe-Descartes-Wotton including central fields whose dynamics is formed by planar affine algebraic curves of degree not exceeding N (N = 1, 2, 3,...).

Geometry of Spectral Curves and All Order Dispersive Integrable System

We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large. The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in 1/N an isomonodromic problem given by a Lax pair, and the relation between "correlators", the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.

Plane algebraic curves with many cusps, with an appendix by Eugenii Shustin

The maximum number $$k(d)$$ of cusps on a plane algebraic curve of degree $$d$$ is an open classical problem that dates back to the nineteenth century. A related open problem is the asymptotic value ( $$a.v.$$ ) of the number of cusps on plane curves, that is $$a.v.=\limsup _{d\rightarrow \infty } k(d)/{d^2}$$ . In this paper, we improve the best known lower bound for the asymptotic value by constructing curves with the largest known number of cusps for infinitely many degrees. Some particular curves of relatively low degree with many cusps are constructed too. The “Appendix” to this paper is devoted to the case of degree 11 and it is due to E. Shustin.

Using a bihomogeneous resultant to find the singularities of rational space curves

We provide a new technique to detect the singularities of rational space curves. Given a rational parametrization of a space curve, we first compute a μ-basis for the parametrization. From this μ-basis we generate three planar algebraic curves of different bidegrees whose intersection points correspond to the parameters of the singularities. To find these intersection points, we construct a new sparse resultant matrix for these three bivariate polynomials. We then compute the parameter values corresponding to the singularities by applying Gaussian elimination to this resultant matrix. Let νQ denote the multiplicity of the singular point Q, and let n be the degree of the curve. We find that when ∑νQ⩽2n−3, the last nonzero row after Gaussian elimination represents a univariate polynomial whose roots are exactly the parameter values of the singularities with the correct multiplicity. Otherwise the last two nonzero rows represent two bivariate polynomials whose common roots provide the parameter values of the singularities. We also show that if R is this resultant matrix, then size(R)−rank(R) gives the total multiplicity ∑νQ(νQ−1) of all the singular points including the infinitely near singular points of a rational space curve and we provide bounds on the expression ∑νQ(νQ−1) for the total multiplicity of all the singular points of a rational space curve. To verify our results, we present several examples to illustrate our methods.

On the size of the automorphism group of a plane algebraic curve

Let K be an algebraically closed field of characteristic p>0, and let X be a curve over K of genus g≥2. Assume that p>2 and that X admits a non-singular plane model. The following result is proven: if X has more than 3(2g2+g)(8g+1+3) automorphisms, then X is birationally equivalent to a Hermitian curve.

An effective “Theorem of André” for CM-points on a plane curve

AbstractIt is a well known result of Y. André (a basic special case of the André-Oort conjecture) that an irreducible algebraic plane curve containing infinitely many points whose coordinates are CM-invariants is either a horizontal or vertical line, or a modular curve Y0(n). André's proof was partially ineffective, due to the use of (Siegel's) class-number estimates. Here we observe that his arguments may be modified to yield an effective proof. For example, with the diagonal line X1+X2=1 or the hyperbola X1X2=1 it may be shown quite quickly that there are no imaginary quadratic τ1,τ2 with j(τ1)+j(τ2)=1 or j(τ1)j(τ2)=1, where j is the classical modular function.

Parametrization of Algebraic Curves from a Number Theorist's Point of View

We present the technique of parametrization of plane algebraic curves from a number theorist's point of view and present Kapferer's simple and beautiful (but little known) proof that nonsingular curves of degree > 2 cannot be parametrized by rational functions.

Homeomorphic approximation of the intersection curve of two rational surfaces

We present an approach of computing the intersection curve C of two rational parametric surface S1(u,s) and S2(v,t), one being projectable and hence can easily be implicitized. Plugging the parametric surface to the implicit surface yields a plane algebraic curve G(v,t)=0. By analyzing the topology graph G of G(v,t)=0 and the singular points on the intersection curve C we associate a space topology graph to C, which is homeomorphic to C and therefore leads us to an approximation for C in a given precision.

Finding the topology of implicitly defined two algebraic plane curves

An algorithm is given for computing in a very efficient way the topology of two real algebraic plane curves defined implicitly. The authors preform a symbolic pre-processing that allows us later to execute all numerical computations in an accurate way.

A guide to plane algebraic curves

This book can be used in a one semester undergraduate course or senior capstone course, or as a useful companion in studying algebraic geometry at the graduate level. This Guide is a friendly introduction to plane algebraic curves. It emphasizes geometry and intuition, and the presentation is kept concrete. You'll find an abundance of pictures and examples to help develop your intuition about the subject, which is so basic to understanding and asking fruitful questions. Highlights of the elementary theory are covered, which for some could be an end in itself, and for others an invitation to investigate further. Proofs, when given, are mostly sketched, some in more detail, but typically with less. References to texts that provide further discussion are often included. Computer algebra software has made getting around in algebraic geometry much easier. Algebraic curves and geometry are now being applied to areas such as cryptography, complexity and coding theory, robotics, biological networks, and coupled dynamical systems. Algebraic curves were used in Andrew Wiles' proof of Fermat's Last Theorem, and to understand string theory, you need to know some algebraic geometry. There are other areas on the horizon for which the concepts and tools of algebraic curves and geometry hold tantalizing promise. This introduction to algebraic curves will be appropriate for a wide segment of scientists and engineers wanting an entrance to this burgeoning subject.

Morse theory for plane algebraic curves

We use Morse theoretical arguments to study algebraic curves in C^2. We take an algebraic curve C in C^2 and intersect it with a family of spheres with fixed origin and varying radii. We explain in detail how does the resulting link change when we cross a singular point of C. Applying link invariants as Murasugi's signature and Levine--Tristram signatures we obtain some informations about possible singularities of a curve C in terms of its topology.

Spectrum of plane curves via knot theory

In this paper, we use topological methods to study various semicontinuity properties of the local spectrum of singular points of algebraic plane curves and spectrum at infinity of polynomial maps in two variables. Using the Seifert form, the Tristram–Levine signatures of links, and the associated Murasugi-type inequalities, we reprove (in a slightly weaker form) a result obtained by Steenbrink and Varchenko on semicontinuity of the spectrum of singular points under deformations and result of Némethi and Sabbah on semicontinuity of the spectrum at infinity regarding families of polynomial maps. We also relate the spectrum at infinity of a polynomial map with the collection of the spectra of singular points of a chosen fiber.

Symbolic-numeric algorithms for plane algebraic curves

Symbolic-numeric algorithms for plane algebraic curves

INFLECTION POINTS OF REAL AND TROPICAL PLANE CURVES

We prove that Viro’s patchworking produces real algebraic curves with the maximal number of real inflection points. In particular this implies that maximally inflected real algebraic M -curves realize many isotopy types. The strategy we adopt in this paper is tropical: we study tropical limits of inflection points of classical plane algebraic curves. The main tropical tool we use to understand these tropical inflection points are tropical modifications.

Irreducibility Criterion For Quasi-Ordinary Polynomials

We give a criterion for a quasi ordinary polynomial to be locally irreducible. The criterion uses the notion of approximate roots and that of generalized Newton polygons. It generalizes the one given by Abhyankar for algebraic plane curves. Introduction Let K be an algebraically closed field of characteristic zero, and let R = K[[x1, . . . , xe]] = K[[x]] be the ring of formal power series in x1, . . . , xe over K. Let f = y+a1(x)y + . . .+an(x) be a nonzero polynomial of R[y], and suppose that f is irreducible in R[y]. Suppose that e = 1 and let g be a nonzero polynomial of R[y], then define the intersection multiplicity of f with g, denoted int(f, g), to be the x-order of the y resultant of f and g. The set of int(f, g), g ∈ R[y], defines a semigroup, denoted Γ(f). It is will known that a set of generators of Γ(f) can be computed from polynomials having the maximal contact with f (see [1] and [6]), namely, there exist g1, . . . , gh such that n, int(f, g1), . . . , int(f, gh) generate Γ(f) and for all 1 ≤ k ≤ h, the Newton-Puiseux expansion of gk coincides with that of f until a characteristic exponent of f . In [1], Abhyankar introduced a special set of polynomials called the approximate roots of f . These polynomials have the advantage that they can be calculated from the equation of f by using the Tschirnhausen transform. Suppose that e ≥ 2 and that the discriminant of f is of the form x1 1 . . . . .x Ne e .u(x1, . . . , xe), where u is a unit in K[[x]] (such a polynomial is called quasi-ordinary polynomial). By Abhyankar-Jung Theorem, the roots of f(x1, . . . , xe, y) = 0 are all in K[[x 1 n 1 , . . . , x 1 n e ]], in particular there exists a power series y(t1, . . . , te) = ∑ p cpt p1 1 . . . . .t pe e ∈ K[[t1, . . . , te]] such that f(t1 , . . . , t n e , y(t1, . . . , te)) = 0 and the other roots of f(t n 1 , . . . , t n e , y) = 0 are the conjugates of y(t1, . . . , te) with respect to the nth roots of unity in K. Given a polynomial g of R[y], we define the order of g to be the leading exponent with respect to the lexicographical order of the smallest homogeneous component of g(t1 , . . . , t n e , y(t1, . . . , te)). The set of orders of polynomials of R[y] defines a semigroup. In this paper we first prove that the canonical basis of (nZ) with the set of orders of the approximate roots of f generate the semigroup of f , then we give, using these approximate roots and the notion of generalized Newton polygons, a criterion for a quasi-ordinary polynomial to be irreducible. Note that if e = 1, then f is quasi-ordinary, in particular our results generalize those of Abhyankar (see [1] and [3]). The paper is organized as follows: in Section 1 we introduce the notion of approximate roots of a polynomial in one variable over a commutative ring with unity. In Section 2 we show how to associate a semigroup with an irreducible quasi-ordinary polynomial of R[y]. In Section 3 we introduce the notion of pseudo roots of a quasi-ordinary polynomial f then we prove that the orders of these polynomials together with the canonical basis of (nZ) give a set of generators of the semigroup of f . This result remains true if we replace the pseudo roots of f by its set of approximate roots. This is what we prove in Section 4. Sections 5 and 6 are devoted to the irreducibility criterion: in Section 5 we introduce the notion of generalized Newton polygon, and we define ∗2000 Mathematical Subject Classification: 32S25, 32S70. During the development of this work, the author visited the Department of Mathematics at the American University of Beirut, Lebanon. He would like to thank that institution for hospitality and support. He also would like to think the Center for Advanced Mathematical Sciences-CAMS for offering access to many facilities. †Universite d’Angers, Mathematiques, 49045 Angers cedex 01, France, e-mail:assi@univ-angers.fr

A worst-case bound for topology computation of algebraic curves

Computing the topology of an algebraic plane curve C means computing a combinatorial graph that is isotopic to C and thus represents its topology in R 2 . We prove that, for a polynomial of degree n with integer coefficients bounded by 2 ρ , the topology of the induced curve can be computed with O ̃ ( n 8 ρ ( n + ρ ) ) bit operations ( O ̃ indicates that we omit logarithmic factors). Our analysis improves the previous best known complexity bounds by a factor of n 2 . The improvement is based on new techniques to compute and refine isolating intervals for the real roots of polynomials, and on the consequent amortized analysis of the critical fibers of the algebraic curve.

On the total order of reducibility of a pencil of algebraic plane curves

In this paper, the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed. Unlike most of the previous related works, each reducible curve of the pencil is here counted with its appropriate multiplicity. It is proved that this number of reducible curves, counted with multiplicity, is bounded by d 2 − 1 where d is the degree of the pencil. Then, a sharper bound is given by taking into account Newtonʼs polygon of the pencil.