Plane Algebraic Curves Research Articles

Complex algebraic plane curves via the Poincaré–Hopf formula, I: Parametric lines

We present a new and very eficient approach to study topology of algebraic curves in C2. It relies on using the Poincare?Hopf formula, applied to a suitable Hamiltonian vector field, to estimate the number of double points and the Milnor numbers of singular points of the curve and on considering finite-dimensional spaces of curves with given asymptotics at infinity. We apply this method to classification of parametric lines with one self-intersection.

A compactness criterion for real plane algebraic curves

Two sets of conditions are presented for the compactness of a real plane algebraic curve, one sufficient and one necessary, in terms of the Newton polygon of the defining polynomial.

Parametrization of algebraic curves defined by sparse equations

We present a new method for the rational parametrization of plane algebraic curves. The algorithm exploits the shape of the Newton polygon of the defining implicit equation and is based on methods of toric geometry.

Analytic equivalence of plane curve singularities yn +xαy +xβa(x) = 0

There are not many examples of complete analytical classification of specific families of singularities, even in the case of plane algebraic curves. In 1989, Kang and Kim published a paper on analytical classification of plane curve singularities yn+a(x)y+b(x) = 0, or, equivalently, yn+x?y+x?A(x) = 0 where A(x) is a unit in Ct{x}, ? and ? are integers, ? _ n ? 1 and ? _ n. The classification was not complete in the most difficult case ? n?1 = ? n. In the present paper, the classification is extended also in this case, the proofs are improved and some gaps are removed.

Curves in Cages: An Algebro-Geometric Zoo

This paper is concerned with families of plane algebraic curves that contain a given and quite special set of points X . We focus on the case in which the set X is formed by transversally intersecting pairs of lines selected from two given finite families. The union of all lines from both families is called a cage (this notion of cage will be made more precise later), and the intersection X consists of points at which a line from the first family intersects a line from the second. The points of X are called the nodes of the cage. This is a particular case of a more general problem. Let X be the intersection set of two plane algebraic curves D and E that do not share a common component, that is, do not contain a common irreducible curve. If d and e denote the degrees of D and E , respectively, then X consists of at most d · e points. When the cardinality of X is exactly d · e, X is called a complete intersection (complete intersections have many nice properties). How does one describe polynomials of degree at most k that vanish on a complete intersection X or on its subsets? This problem has a glorious history (see [1], [2], or [3]) and its generalizations are a subject of active and exciting research [4], [5], [6], [7], [8], [9]. When k is much larger then d · e, points of X impose independent restrictions on polynomials of degree k. However, for small k these restrictions fail to be independent. In such cases, we can look for maximal subsets of X that impose independent constraints. In this article, we deal with the special case of this classical problem suggested earlier, namely, the case in which both plane curves D and E are simply unions of lines and the union D ∪ E is the (d× e)-cage in question. Note that D is the zero set of a product of d linear polynomials, while E is the zero set of a product of e linear polynomials. Hence, the degrees of D and E are d and e, respectively. In order to simplify our terminology, we color the lines from D red and the ones from E blue.

Curves in Cages: An Algebro-Geometric Zoo

The paper is concerned with families of plane algebraic curves that contain a given and quite special finite set X of points in the projective plane. We focus on the case in which the set X is formed by transversally intersecting pairs of lines selected from two given finite families of cardinality d. The union of all lines from both families is called a cage, and the intersection X consists of d^2 points at which a line from the first family intersects a line from the second. The points of X are called the nodes of the cage. We study the subsets A of the nodal set X such that any plane algebraic curve C of degree d that contains A must contain X as well. As a corollary, we get a few generalizations of the famous Pascal theorem, generalizations that employ polygons (instead of hexagons as in Pascal's theorem) inscribed in a quadratic curve. Our results are closely related to the classical theorems of Chasles and Bacharach. Although the nodal sets X produced by our cages are quite special in comparison to more general complete intersections studied by Bacharach, the cages provide us with a much better grip on the combinatorics of subsets A of X with the property that is described above.

Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme

We describe a method that serves to simultaneously determine the topological configuration of the intersection curve of two parametric surfaces and generate compatible decompositions of their parameter domains, that are amenable to the application of existing perturbation schemes ensuring exact topological consistency of the trimmed surface representations. To illustrate this method, we begin with the simpler problem of topology resolution for a planar algebraic curve F(x,y)=0 in a given domain, and then extend concepts developed in this context to address the intersection of two tensor-product parametric surfaces p(s,t) and q(u,v) defined on (s,t)∈[0,1]2 and (u,v)∈[0,1]2. The algorithms assume the ability to compute, to any specified precision, the real solutions of systems of polynomial equations in at most four variables within rectangular domains, and proofs for the correctness of the algorithms under this assumption are given.

Robust Plotting of Polar Algebraic Curves, Space Algebraic Curves, and Offsets of Planar Algebraic Curves

In this paper, based on modifications to the well-known subdivision scheme in combination with modified affine arithmetic method to guide the subdivision, we propose four robust and reliable algorithms for plotting polar algebraic curves, space algebraic curves and offsets of planar algebraic curves.

Determination of the tangents for a real plane algebraic curve

The purpose of this paper is to present an algorithm for computing the tangents of a real plane algebraic curve. By this algorithm, all the slopes of the tangents to a real plane algebraic curve at a particular point may be accurately represented via polynomial real root isolation.

A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs

We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For a first order autonomous ODE F = 0 , we give an exact degree bound for its rational solutions, based on the connection between rational solutions of F = 0 and rational parametrizations of the plane algebraic curve defined by F = 0 . For a first order autonomous ODE, we further give a polynomial time algorithm for computing a rational general solution if it exists based on the computation of Laurent series solutions and Padé approximants. Experimental results show that the algorithm is quite efficient.

Dimers and amoebae

We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on a weighted, bipartite, doubly periodic graph G embedded in the plane. We derive explicit formulas for the surface tension and local Gibbs measure probabilities of these models. The answers involve a certain plane algebraic curve, which is the spectral curve of the Kasteleyn operator of the graph. For example, the surface tension is the Legendre dual of the Ronkin function of the spectral curve. The amoeba of the spectral curve represents the phase diagram of the dimer model. Further, we prove that the spectral curve of a dimer model is always a real curve of special type, namely it is a Harnack curve. This implies many qualitative and quantitative statement about the behavior of the dimer model, such as existence of smooth phases, decay rate of correlations, growth rate of height function fluctuations, etc.

The Automorphism Group of Plane Algebraic Curves with Singer Automorphisms

The main result in Cossidente and Siciliano (J. Number Theory, Vol. 99 (2003) pp. 373---382) states that if a Singer subgroup of PGL(3,q) is an automorphism group of a projective, geometric irreducible, non-singular plane algebraic curve $$\mathcal{X}$$ then either $$\deg(\mathcal{X})=q+2$$ or $$\deg(\mathcal {X})\ge q^2+q+1$$ . In the former case $$\mathcal{X}$$ is projectively equivalent to the curve $$\mathcal{X}_q$$ with equation Xq+1Y+Yq+1+X=0 studied by Pellikaan. Furthermore, the curve $$\mathcal{X}_q$$ has a very nice property from Finite Geometry point of view: apart from the three distinguished points fixed by the Singer subgroup, the set of its $$\mathbb{F}_{{q}^{3}}$$ -rational points can be partitioned into finite projective planes $$P^{2}(\mathbb{F}_{q})$$ . In this paper, the full automorphism group of such curves is determined. It turns out that $$Aut(\mathcal {X}_q)$$ is the normalizer of a Singer group in $$PGL(3,\mathbb{F}_{q})$$ .

A COMMENT ON A NONIDEAL CENTRIFUGAL VIBRATOR MACHINE BEHAVIOR WITH SOFT AND HARD SPRINGS

This paper concerns a type of rotating machine (centrifugal vibrator), which is supported on a nonlinear spring. This is a nonideal kind of mechanical system. The goal of the present work is to show the striking differences between the cases where we take into account soft and hard spring types. For soft spring, we prove the existence of homoclinic chaos. By using the Melnikov's Method, we show the existence of an interval with the following property: if a certain parameter belongs to this interval, then we have chaotic behavior; otherwise, this does not happen. Furthermore, if we use an appropriate damping coefficient, the chaotic behavior can be avoided. For hard spring, we prove the existence of Hopf's Bifurcation, by using reduction to Center Manifolds and the Bezout Theorem (a classical result about algebraic plane curves).

Real plane algebraic curves with asymptotically maximal number of even ovals

It has been known for a long time that a nonsingular real algebraic curve of degree 2k in the projective plane cannot have more than 7k2/4-9k/4+3/2 even ovals. We show here that this upper bound is asymptotically sharp; that is to say, we construct a family of curves of degree 2k such that p/k2→k→∞7/4, where p is the number of even ovals of the curves. We also show that the same kind of result is valid when dealing with odd ovals

Higher-order Alexander invariants of plane algebraic curves

We define new higher-order Alexander modules An(C) and higher-order degrees δn(C) which are invariants of the algebraic planar curve C. These come from analyzing the module structure of the homology of certain solvable covers of the complement of the curve C. These invariants are in the spirit of those developed by Cochran and Harvey, which were used to study knots, 3-manifolds, and finitely presented groups, respectively. We show that for curves in general position at infinity, the higher-order degrees are finite. This provides new obstructions on the type of groups that can arise as fundamental groups of complements to affine curves in general position at infinity.

Global B-Spline Approximations of Singular Real Algebraic Plane Curves

This paper proposes an algorithm for globally approximating real algebraic plane curves of degree k with B-spline curves of the same degree. Each connected component is approximated with a B-spline curve. It is suitable for all irreducible real plane algebraic curves with arbitrary genus (including singular curves). This method is based on our blowup sampling method of algebraic curves, which solves the difficult problem of sampling around singular points in essence. The experimental results show that the algorithm achieves better accuracy than the

Holomorphic Dynamics in Loop Spaces

We discuss a certain type of evolution of loops in smooth manifolds described in terms of natural almost complex structures on appropriate loop spaces. The main attention is paid to holomorphic families of transverse loops in a Cauchy---Riemann manifold, with an emphasis on the geometry of their images. In this context, we introduce a special class of foliated solid tori, called holomorphic doughnuts, and show that they are connected with certain classical geometric constructions. In particular, we show that holomorphic doughnuts can be obtained from isolated singularities of algebraic plane curves. We also establish the existence of holomorphic dynamics for generic real-analytic loops.

Exact, efficient, and complete arrangement computation for cubic curves

The Bentley–Ottmann sweep-line method can compute the arrangement of planar curves, provided a number of geometric primitives operating on the curves are available. We discuss the reduction of the primitives to the analysis of curves and curve pairs, and describe efficient realizations of these analyses for planar algebraic curves of degree three or less. We obtain a complete, exact, and efficient algorithm for computing arrangements of cubic curves. Special cases of cubic curves are conics as well as implicitized cubic splines and Bézier curves. The algorithm is complete in that it handles all possible degeneracies such as tangential intersections and singularities. It is exact in that it provides the mathematically correct result. It is efficient in that it can handle hundreds of curves with a quarter million of segments in the final arrangement. The algorithm has been implemented in C ++ as an Exacus library called CubiX.

Some constructions for real algebraic curves

We address two basic questions for real algebraic curves. The first one is how to decide whether a real algebraic curve in the n -projective space contains some real point. We present an algorithm that reduces the original question to deciding whether the zero-set of a zero-dimensional ideal contains real points. The second part of the paper is devoted to giving necessary and sufficient conditions for the existence of a real line disjoint from a given real plane algebraic curve. An algorithm for testing whether these conditions are fulfilled is given.

Holomorphic tubes in Cauchy–Riemann manifolds†

We consider pseudoholomorphic curves in loop spaces of Cauchy–Riemann manifolds and relate them to certain geometric and topological constructions. Specifically, we discuss pseudoholomorphic functions of one complex variable taking values in the space of transverse loops in a Cauchy–Riemann manifold, with an emphasis on the geometry of their images. In this context, we introduce a special class of foliated solid tori, called holomorphic tubes, which appear connected with a number of interesting geometric topics. In particular, we prove that each isolated singularity of algebraic plane curve defines a holomorphic tube in a small 3-sphere around the singularity endowed with the standard CR-structure. We also show that there exist a plenty of holomorphic tubes containing a given real analytic transverse loop. In particular, such a loop can be holomorphically deformed along certain vector fields on its image.