Plane Algebraic Curves Research Articles

On the Jacobian scheme of a plane curve

We study the Jacobian scheme of a plane algebraic curve at an ordinary singularity, characterizing it through a geometric property. We give an algorithm that gives the analytic type of a double point using the algebraic version of the Mather-Yau Theorem. We give a characterization of plane nodal curves with no infinitely near points through their Tjurina number.

Fundamental group of complex hypocycloids

The Zariski–van Kampen theorem allows us to provide a presentation of the fundamental group for the complement of algebraic plane curves. However, certain computations require arduous work, as exemplified in the case of hypocycloids. In this paper we present the following result: Theorem 1.The fundamental group of any complex hypocycloid withNcusps is the Artin group of theN-gon. The main idea of the proof is take advantage of the symmetries inherent in the hypocycloid, allowing us to partition the domain to determine the generators of the fundamental group.

On G2 approximation of planar algebraic curves under certified error control by quintic Pythagorean-hodograph splines

The Pythagorean-Hodograph curve (PH curve) is a valuable curve type extensively utilized in computer-aided geometric design and manufacturing. This paper presents an approach to approximate a planar algebraic curve within a bounding box by employing piecewise quintic PH spline curves, while maintaining G2 smoothness of the approximating curve and preserving second-order geometric details at singularities. The bounding box encompasses all x-coordinates of key topological points, ensuring accurate representation. The paper explores the analysis of the G2 interpolation problem for quintic PH curves with invariant convexity, transforming the quest for interpolation solutions into identifying positive roots within a set of algebraic equations. Through infinitesimal order analysis, it is established that a solution necessarily exists following adequate subdivision, laying the groundwork for practical application. Finally, the paper introduces a novel algorithm that integrates prior research to construct the approximating curve while maintaining control over the desired error levels.

A survey on recent advances in the computation of asymptotes

AbstractIn this paper, we summarize two algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined. The approach is based on the notion of perfect curve introduced from the concepts and results presented in Blasco and Pérez-Díaz (Comput Aided Geom Des 31(2):81–96, 2014), Blasco and Pérez-Díaz (Comput Aided Geom Des 31(7-8):345–357, 2014), and Blasco and Pérez-Díaz (J Comput Appl Math 278:231–247, 2015). From these results, we derive a new method that allow to easily compute horizontal and vertical asymptotes.

Bounds for Rational Points on Algebraic Curves, Optimal in the Degree, and Dimension Growth

Abstract Bounding the number of rational points of height at most $H$ on irreducible algebraic plane curves of degree $d$ has been an intense topic of investigation since the work by Bombieri and Pila. In this paper we establish optimal dependence on $d$ by showing the upper bound $C d^{2} H^{2/d} (\log H)^{\kappa }$ with some absolute constants $C$ and $\kappa $. This bound is optimal with respect to both $d$ and $H$, except for the constants $C$ and $\kappa $. This answers a question raised by Salberger, leading to a simplified proof of his results on the uniform dimension growth conjectures of Heath-Brown and Serre, and where at the same time we replace the $H^{\varepsilon }$ factor by a power of $\log H$. The main strength of our approach comes from the combination of a new, efficient form of smooth parametrizations of algebraic curves with a century-old criterion of Pólya, which allows us to save one extra power of $d$ compared with the standard approach using Bézout’s theorem.

Expressive curves

We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve C C is expressive if (a) each irreducible component of C C can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of C C in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of C C in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)–(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.

An effective algorithm for computing the asymptotes of an implicit curve

In this paper, we first summarize the algorithm presented in Blasco and Pérez-Díaz (2014) for computing the generalized asymptotes of algebraic curves implicitly defined. This algorithm is based on the computation of Puiseux series. The main and very important contribution of this paper is a new and efficient method that allows to easily compute all the generalized asymptotes of an algebraic plane curve implicitly defined by just solving a triangular system of equations. The method can be easily generalized to the case of algebraic curves implicitly defined in the n-dimensional space.

Design and implementation of symbolic algorithms for the computation of generalized asymptotes

In this paper we present two algorithms for computing the g-asymptotes or generalized asymptotes, of a plane algebraic curve, mathscr {C}, implicitly or parametrically defined. The asymptotes of a curve mathscr {C} reflect the status of mathscr {C} at points with sufficiently large coordinates. It is well known that an asymptote of a curve mathscr {C} is a line such that the distance between mathscr {C} and the line approaches zero as they tend to infinity. However, a curve mathscr {C} may have more general curves than lines describing the status of mathscr {C} at infinity. These curves are known as g-asymptotes or generalized asymptotes. The pseudocodes of these algorithms are presented, as well as the corresponding implementations. For this purpose, we use the algebra software Maple. A comparative analysis of the algorithms is carried out, based on some properties of the input curves and their results to analyze the efficiency of the algorithms and to establish comparative criteria. The results presented in this paper are a starting point to generalize this study to surfaces or to curves defined by a non-rational parametrization, as well as to improve the efficiency of the algorithms. Additionally, the methods developed can provide a new and different approach in prediction (regression) or classification algorithms in the machine learning field.

Rational points close to non‐singular algebraic curves

AbstractWe study the density of solutions to Diophantine inequalities involving non‐singular ternary forms, or equivalently, the density of rational points close to non‐singular plane algebraic curves.

Globally certified [formula omitted] approximation of planar algebraic curves

Topology analysis and geometric feature determination for curves and surfaces are basic problems in geometric modeling and computer aided design. When the curves and surfaces are given in implicit expressions, the problems become more difficult. This paper presents a complete algorithm framework for the problem of certified G1 approximation of a planar algebraic curve under global error control. Compared to other work on topology and geometric analysis of algebraic curves, this method presents a new topology analysis as well as a geometric information computation. Moreover, this method uses generalized asymptotes as the approximation of the unbounded part, which makes it achieve global approximation with given precision.

Poincaré–Reeb graphs of real algebraic domains

An algebraic domain is a closed topological subsurface of a real affine plane whose boundary consists of disjoint smooth connected components of real algebraic plane curves. We study the geometric shape of an algebraic domain by collapsing all vertical segments contained in it: this yields a Poincaré–Reeb graph, which is naturally transversal to the foliation by vertical lines. We show that any transversal graph whose vertices have only valencies 1 and 3 and are situated on distinct vertical lines can be realized as a Poincaré–Reeb graph.

The truncated moment problem on curves y = q(x) and yx ℓ = 1

In this paper, we study the bivariate truncated moment problem (TMP) on curves of the form y = q ( x ) , q ( x ) ∈ R [ x ] , deg ⁡ q ≥ 3 and y x ℓ = 1 , ℓ ∈ N ∖ { 1 } . For even degree sequences, the solution based on the size of moment matrix extensions was first given by Fialkow [Fialkow L. Solution of the truncated moment problem with variety y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.] using the truncated Riesz–Haviland theorem [Curto R, Fialkow L. An analogue of the Riesz–Haviland theorem for the truncated moment problem. J Funct Anal. 2008;255:2709–2731.] and a sum-of-squares representations for polynomials, strictly positive on such curves [Fialkow L. Solution of the truncated moment problem with variety y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.; Stochel J. Solving the truncated moment problem solves the moment problem. Glasgow J Math. 2001;43:335–341.]. Namely, the upper bound on this size is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique, introduced in [Zalar A. The truncated Hamburger moment problem with gaps in the index set. Integral Equ Oper Theory. 2021;93:36.doi: 10.1007/s00020-021-02628-6.; Zalar A. The truncated moment problem on the union of parallel lines. Linear Algebra Appl. 2022;649:186–239. http://doi.org/10.1016/j.laa.2022.05.008.; Zalar A. The strong truncated Hamburger moment problem with and without gaps. J Math Anal Appl. 2022;516:126563. doi: 10.1016/j.jmaa.2022.126563.], and improve Fialkow's bound to deg ⁡ q − 1 (resp. ℓ + 1 ) for curves y = q ( x ) (resp. y x ℓ = 1 ). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are kdeg ⁡ q (resp. k ( ℓ + 1 ) ) for curves y = q ( x ) (resp. y x ℓ = 1 ) for even degree sequences, while for odd ones they are kdeg ⁡ q − ⌈ deg ⁡ q 2 ⌉ (resp. k ( ℓ + 1 ) − ⌊ ℓ 2 ⌋ + 1 ) for curves y = q ( x ) (resp. y x ℓ = 1 ). In the even case, these are counterparts to the result by Riener and Schweighofer [Riener C, Schweighofer M. Optimization approaches to quadrature:a new characterization of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions. J Complex. 2018;45:22–54., Corollary 7.8], which gives the same bound for odd degree sequences on all plane curves. In the odd case, their bound is slightly improved on the curves we study. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases to the TMP on curves y = x ℓ , ℓ = 2 , 3 , and add a new solvability condition to the even degree case on the curve y = x 2 .

The Newton–Puiseux Algorithm and Triple Points for Plane Curves

The paper is an introduction to the use of the classical Newton–Puiseux procedure, oriented towards an algorithmic description of it. This procedure allows to obtain polynomial approximations for parameterizations of branches of an algebraic plane curve at a singular point. We look for an approach that can be easily grasped and is almost self-contained. We illustrate the use of the algorithm first in a completely worked out example of a curve with a point of multiplicity 6, and secondly, in the study of triple points on reduced plane curves.

Orthogonal polynomials on a class of planar algebraic curves

AbstractWe construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form in where and ϕ is a polynomial of arbitrary degree d, in terms of univariate semiclassical OPs. We compute connection coefficients that relate the bivariate OPs to a polynomial basis that is itself orthogonal and whose span contains the OPs as a subspace. The connection matrix is shown to be banded and the connection coefficients and Jacobi matrices for OPs of degree are computed via the Lanczos algorithm in operations.

Computing branches and asymptotes of meromorphic functions

In this paper, we first summarize the existing algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined. From these previous results, we derive a method that allows to easily compute the whole branch and all the generalized asymptotes of a “special” curve defined in n-dimensional space by a parametrization that is not necessarily rational. So, some new concepts and methods are established for this type of curves. The approach is based on the notion of perfect curves introduced from the concepts and results presented in previous papers.

F. S. Macaulay: From plane curves to Gorenstein rings

Francis Sowerby Macaulay began his career working on Brill and Noether’s theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation. Though he never spoke of the connection, the threads of Macaulay’s work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected.

ПОВЕРХНОСТИ ДИАГОНАЛЬНОГО ПЕРЕНОСА ВЕЛАРОИДАЛЬНОГО ТИПА НА РОМБИЧЕСКОМ ПЛАНЕ

The paper illustrates the application of the formulae of general type derived by the author earlier for the definition of surfaces of diagonal translation of superellipses of variable curvature on a rhombic plane. Explicit and parametric equations were derived additionally for the large group of surfaces of diagonal translation of congruent superellipses. For the both cases, the surfaces of velaroidal type are examined on rhombic plane. All of presented surfaces were visualized with the help of methods of computer graphics. Due to availability of arbitrary exponents of powers in explicit equations of generatrix superellipses of the main frame of a translation surface, design of surfaces of diagonal translation was broadened for the case of using plane algebraic curves instead of superellipses in the process of choice of main frame of projected surface of diagonal translation. The presented surfaces can find the application in architecture, civili engineering, and in machine building.

Conductors of Abhyankar-Moh semigroups of even degrees

<abstract><p>In their paper on the embeddings of the line in the plane, Abhyankar and Moh proved an important inequality, now known as the Abhyankar-Moh inequality, which can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve. Barrolleta, García Barroso and Płoski studied the semigroups of integers satisfying the Abhyankar-Moh inequality and call them Abhyankar-Moh semigroups. They described such semigroups with the maximum conductor. In this paper we prove that all possible conductor values are achieved for the Abhyankar-Moh semigroups of even degree. Our proof is constructive, explicitly describing families that achieve a given value as its conductor.</p></abstract>

Some New Symbolic Algorithms for the Computation of Generalized Asymptotes

We present symbolic algorithms for computing the g-asymptotes, or generalized asymptotes, of a plane algebraic curve, C, implicitly or parametrically defined. The g-asymptotes generalize the classical concept of asymptotes of a plane algebraic curve. Both notions have been previously studied for analyzing the geometry and topology of a curve at infinity points, as well as to detect the symmetries that can occur in coordinates far from the origin. Thus, based on this research, and in order to solve practical problems in the fields of science and engineering, we present the pseudocodes and implementations of algorithms based on the Puiseux series expansion to construct the g-asymptotes of a plane algebraic curve, implicitly or parametrically defined. Additionally, we propose some new symbolic methods and their corresponding implementations which improve the efficiency of the preceding. These new methods are based on the computation of limits and derivatives; they show higher computational performance, demanding fewer hardware resources and system requirements, as well as reducing computer overload. Finally, as a novelty in this research area, a comparative analysis for all the algorithms is carried out, considering the properties of the input curves and their outcomes, to analyze their efficiency and to establish comparative criteria between them.

On strongly walk regular graphs, triple sum sets and their codes

Strongly walk regular graphs (SWRGs or s-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length 2 are replaced by paths of length s. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an s-SWRG has at most 4 distinct eigenvalues k> theta _1> theta _2 > theta _3, and that the triple (theta _1, theta _2, theta _3) satisfies a certain homogeneous polynomial equation of degree s - 2 (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for s = 5 and s = 7, there are only the obvious solutions, and we conjecture this to remain true for all (odd) s ge 9.