Rational Plane Curves Research Articles

Enumeration of non-nodal real plane rational curves

Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal singularities and passing through a generic conjugation-invariant configuration of appropriately many points in the plane. We show that an invariant like this is unique: it enumerates real rational three-cuspidal quartics that pass through generically chosen four pairs of complex conjugate points. As a consequence, we show that through any generic configuration of four pairs of complex conjugate points, one can always trace a pair of real rational three-cuspidal quartics.

Detecting similarities of rational plane curves using complex differential invariants

We present a new and efficient method to detect whether or not two given rational plane curves are similar. If both curves are the same, the method finds the symmetries of the curve. The method relies on the introduction of a complex differential invariant that has a nice behavior with respect to Möbius transformations, which are the mappings lying behind the similarities in the parameter space. From a computational point of view, our algorithm only requires bivariate gcds and factoring. The algorithm is implemented in Maple™ [Maplesoft, a division of Waterloo Maple Inc. Waterloo, Ontario (2021)]. An extensive study of its practical performance is also provided.

Rodrigues’ Descendants of a Polynomial and Boutroux Curves

Motivated by the classical Rodrigues’ formula, we study below the root asymptotic of the polynomial sequence R[αn],n,P(z)=d[αn]Pn(z)dz[αn],n=0,1,⋯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathcal {R}}_{[\\alpha n],n,P}(z)=\\frac{\\mathop {}\\!\ extrm{d}^{[\\alpha n]}P^n(z)}{\\mathop {}\\!\ extrm{d}z^{[\\alpha n]}}, n= 0,1,\\dots \\end{aligned}$$\\end{document}where P(z) is a fixed univariate polynomial, α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is a fixed positive number smaller than degP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\deg P$$\\end{document}, and [αn]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[\\alpha n]$$\\end{document} stands for the integer part of αn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha n$$\\end{document}. Our description of this asymptotic is expressed in terms of an explicit harmonic function uniquely determined by the plane rational curve emerging from the application of the saddle point method to the integral representation of the latter polynomials using Cauchy’s formula for higher derivatives. As a consequence of our method, we conclude that this curve is birationally equivalent to the zero locus of the bivariate algebraic equation satisfied by the Cauchy transform of the asymptotic root-counting measure for the latter polynomial sequence. We show that this harmonic function is also associated with an abelian differential having only purely imaginary periods and the latter plane curve belongs to the class of Boutroux curves initially introduced in Bertola (Anal Math Phys 1: 167–211, 2011), Bertola and Mo (Adv Math 220(1): 154–218, 2009). As an additional relevant piece of information, we derive a linear ordinary differential equation satisfied by {R[αn],n,P(z)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{{\\mathcal {R}}_{[\\alpha n],n,P}(z)\\}$$\\end{document} as well as higher derivatives of powers of more general functions.

Remarks on double points of plane curves

We study the relation between the type of a double point of a plane curve and the curvilinear 0-dimensional subschemes of the curve at the point. An Algorithm related to a classical procedure for the study of double points via osculating curves is described and proved. Eventually we look for a way to create examples of rational plane curves with given singularities A_s.

On a problem of Lang for matrix polynomials

In this paper, we consider a problem of Lang about finiteness of torsion points on plane rational curves, and prove some results towards a matrix analogue of this problem, including a full analogue for 2 × 2 $2\times 2$ matrices defined over C $\mathbb {C}$ .

Counting planar curves in [formula omitted] with degenerate singularities

In this paper, we consider the following question: how many degree d curves are there in P3 (passing through the right number of generic lines and points), whose image lies inside a P2, having δ nodes and one singularity of codimension k? We obtain an explicit formula for this number when δ+k≤4 (i.e. the total codimension of the singularities is not more than four). We use a topological method to compute the degenerate contribution to the Euler class; it is an extension of the method that originates in the paper by A. Zinger [30] and which is further pursued by S. Basu and the second author in [1], [2] and [3]. Using this method, we have obtained formulas when the singularities present are more degenerate than nodes (such as cusps, tacnodes and triple points). When the singularities are only nodes, we have verified that our answers are consistent with those obtained by S. Kleiman and R. Piene (in [17]) and by T. Laarakker (in [19]). We also verify that our answer for the characteristic number of planar cubics with a cusp and the number of planar quartics with two nodes and one cusp is consistent with the answer obtained by R. Singh and the second author (in [22]), where they compute the characteristic number of rational planar curves in P3 with a cusp. We also verify some of the numbers predicted by the conjecture made by Pandharipande in [23], regarding the enumerativity of BPS numbers for P3.

Sheaf Theoretic Compactifications of the Space of Rational Quartic Plane Curves

Let $R_4$ be the space of rational plane curves of degree $4$. In this paper, we obtain a sheaf theoretic compactification of $R_4$ via the space of $\alpha$-semistable pairs on $\mathbb{P}^2$ and its birational relations through wall-crossings of semistable pairs. We obtain the Poincare polynomial of the compactified space.

Computing μ-Bases of Univariate Polynomial Matrices Using Polynomial Matrix Factorization

This paper extends the notion of μ-bases to arbitrary univariate polynomial matrices and present an efficient algorithm to compute a μ-basis for a univariate polynomial matrix based on polynomial matrix factorization. Particularly, when applied to polynomial vectors, the algorithm computes a μ-basis of a rational space curve in arbitrary dimension. The authors perform theoretical complexity analysis in this situation and show that the computational complexity of the algorithm is $${\cal O}\left( {d{n^4} + {d^2}{n^3}} \right)$$ , where n is the dimension of the polynomial vector and d is the maximum degree of the polynomials in the vector. In general, the algorithm is n times faster than Song and Goldman’s method, and is more efficient than Hoon Hong’s method when d is relatively large with respect to n. Especially, for computing μ-bases of planar rational curves, the algorithm is among the two fastest algorithms.

Inversion, degree, reparametrization and implicitization of improperly parametrized planar curves using μ-basis

The μ-basis of a rational curve/surface is a new algebraic tool which plays an important role in connecting the rational parametric form and the implicit form of a rational curve/surface. However, most results for μ-bases are presented for proper rational parametrizations. In this paper we consider the μ-basis for an improper rational planar curve. Based on the known properties and new results, we design two new proper reparametrization algorithms using μ-basis. The inversion, degree of the induced rational map and implicitization formulas are also derived.

On refined count of rational tropical curves

We address the problem of existence of refined (i.e., depending on a formal parameter) tropical enumerative invariants, and we present two new examples of a refined count of rational marked tropical curves. One of the new invariants counts plane rational tropical curves with an unmarked vertex of arbitrary valency. It was motivated by the tropical enumeration of plane cuspidal tropical curves given by Y. Ganor and the author, which naturally led to consideration of plane tropical curves with an unmarked four-valent vertex. Another refined invariant counts rational tropical curves of a given degree in the Euclidean space of arbitrary dimension matching specific constraints, which make the spacial refined invariant similar to known planar invariants.

Singular Welschinger invariants

We suggest an invariant way to enumerate nodal and nodal-cuspidal real deformations of real plane curve singularities. The key idea is to assign Welschinger signs to the counted deformations. Our invariants can be viewed as a local version of Welschinger invariants enumerating real plane rational curves.

Freeness and invariants of rational plane curves

Given a parameterization $\phi$ of a rational plane curve C, we study some invariants of C via $\phi$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via $\phi$, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via $\phi$, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a rational curve can be computed directly from one of its parameterization, without relying on the computation of an equation of C.

ISSAC 2019 software presentations communicated by Yue Ren

Implicit is a package for implicitizing rational planar curves and rational tensor product surfaces, developed on Maplesoft based on the state-of-the-art implicitization techniques. The main functions ImpCurve and ImpSurface return the implicit equation of a rational planar curve or a rational tensor product surface. Other popularly used functions, such as ImpDegree, ImpMatrix and ImpRuled that are used for deciding the implicit degree and the implicit matrix of a general rational surface, and computing the implicit equation of a rational ruled surface in a more efficient way, are also proposed.

Plane curves with three syzygies, minimal Tjurina curves, and nearly cuspidal curves

We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom–Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of curves.

Representing rational curve segments and surface patches using semi-algebraic sets

We provide a framework for representing segments of rational planar curves or patches of rational tensor product surfaces with no singularities using semi-algebraic sets. Given a rational planar curve segment or a rational tensor product surface patch with no singularities, we find the implicit equation of the corresponding unbounded curve or surface and then construct an algebraic box defined by some additional equations and inequalities associated to the implicit equation. This algebraic box is proved to include only the given curve segment or surface patch without any extraneous parts of the unbounded curve or surface. We also explain why it is difficult to construct such an algebraic box if the curve segment or surface patch includes some singular points such as self-intersections. In this case, we show how to isolate a neighborhood of these special points from the corresponding curve segment or surface patch and to represent these special points with small curve segments or surface patches. This framework allows us to dispense with expensive approximation methods such as voxels for representing surface patches.

The singular points of a rational plane curve and its restricted tangent bundle

We consider the vector bundle φ⁎TP2 where φ denotes a generically one-to-one parametrization of a rational curve D of degree dD≥2 in P2. We study a parameter aD that controls the splitting of φ⁎TP2 as a direct sum of line bundles. We find that specific properties of D, related to its singular points, characterize aD. These properties are: multiplicity, non-collinearity of triples, and construction of a curve of degree strictly smaller than dD with multiplicity bounded by the multiplicity of D at corresponding points.

The square-freeness of the offset equation to a rational planar curve, computed via resultants

It is well known [Algebraic properties of plane offset curves, Comput. Aided Geom. Des. 7 (1990) 101–127] that an implicit equation of the offset to a rational planar curve can be computed by removing the extraneous components of the resultant of two certain polynomials computed from the parametrization of the curve. Furthermore, it is also well known that the implicit equation provided by the nonextraneous component of this resultant has at most two irreducible factors [Algebraic analysis of offsets to hypersurfaces, Math. Z. 234 (2000) 697–719]. In this paper, we complete the algebraic description of this resultant by showing that the multiplicity of the factors corresponding to the offset can be computed in advance. In particular, when the parametrization is proper, i.e. when the curve is just traced once by the parametrization, we prove that any factor corresponding to a simple component of the offset has multiplicity 1, while the factor corresponding to the special component, if any, has multiplicity 2. Hence, if the parametrization is proper and there is no special component, the nonextraneous part of the resultant is square-free. In fact, this condition is proven to be also sufficient. Therefore, this result provides a simple test to check whether or not the offset of a given rational curve has a special component, and in turn, whether a given rational curve is the offset of another curve.

Rational curves over generalized complex numbers

Complex rational curves have been used to represent circular splines as well as many classical curves including epicycloids, cardioids, Joukowski profiles, and the lemniscate of Bernoulli. Complex rational curves are known to have low degree (typically half the degree of the corresponding rational planar curve), circular precision, invariance with respect to Möbius transformations, special implicit forms, an easy detection procedure, and a fast algorithm for computing their μ-bases. But only certain very special rational planar curves are also complex rational curves. To construct a wider collection of curves with similar appealing properties, we generalize complex rational curves to hyperbolic and parabolic rational curves by invoking the hyperbolic and parabolic numbers. We show that the special properties of complex rational curves extend to these hyperbolic and parabolic rational curves. We also provide examples to flesh out the theory.

Classification of planar rational cuspidal curves I. C∗∗-fibrations

To classify complex rational cuspidal curves $E\subseteq \mathbb{P}^2$ it remains to classify the ones with complement of log general type, i.e. the ones for which $\kappa(K_X+D)=2$, where $(X,D)$ is a log resolution of $(\mathbb{P}^2,E)$. It is conjectured that $\kappa(K_X+\frac{1}{2}D)=-\infty$ and hence $\mathbb{P}^2\setminus E$ is $\mathbb{C}^{**}$-fibered, where $\mathbb{C}^{**}=\mathbb{C}^1\setminus\{0,1\}$, or $-(K_X+\frac{1}{2}D)$ is ample on some minimal model of $(X,\frac{1}{2}D)$. Here we classify, up to a projective equivalence, those rational cuspidal curves for which the complement is $\mathbb{C}^{**}$-fibered. From the rich list of known examples only very few are not of this type. We also discover a new infinite family of bicuspidal curves with unusual properties.

Singularities of plane rational curves via projections

We consider the parameterization f=(f0:f1:f2) of a plane rational curve C of degree n, and we study the singularities of C via such parameterization. We use the projection from the rational normal curve Cn⊂Pn to C and its interplay with the secant varieties to Cn. In particular, we define via f certain 0-dimensional schemes Xk⊂Pk, 2≤k≤(n−1), which encode all information on the singularities of multiplicity ≥k of C (e.g. using X2 we can give a criterion to determine whether C is a cuspidal curve or has only ordinary singularities). We give a series of algorithms which allow one to obtain information about the singularities from such schemes.