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For a point $p\in CP^2$ and a triple $(g,d,\ell)$ of non-negative integers we define a {\em Hurwitz--Severi number} ${\mathfrak H}_{g,d,\ell}$ as the number of generic irreducible plane curves of genus $g$ and degree $d+\ell$ having an $\ell$-fold node at $p$ and at most ordinary nodes as singularities at the other points, such that the projection of the curve from $p$ has a prescribed set of local and remote tangents and lines passing through nodes. In the cases $d+\ell\ge g+2$ and $d+2\ell \ge g+2 > d+\ell$ we express the Hurwitz--Severi numbers via appropriate ordinary Hurwitz numbers. The remaining case $d+2\ell, Comment: Version 2: title changed and some explanations added

In this note we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and Tjurina numbers of an isolated plane curve singularity in the cases of one Puiseux pair and semi-quasi-homogeneous singularities., To Appear in Comptes Rendus Mathematique, Ser. I 357 (2019), 205-208

The idea of envelope of a family of plane curves is an elementary notion in differential geometry. As such, its implementation in dynamic geometry environments is quite universal (Cabri, The Geometer’s Sketchpad, Cinderella, GeoGebra,...). Nevertheless, most of these programs return, when computing certain envelopes, both some spurious solutions and the curves that truly fit in the intuitive definition of envelope. The precise distinction between spurious and genuine parts has not been made before: This paper proposes such distinction in an algorithmic way, ready for its implementation in interactive geometry systems, allowing a finer classification of the different parts resulting from the current, advanced approach to envelope computation and, thus, yielding a more precise output, free from extraneous components.

F. Cukierman asked whether or not for every smooth real plane curve $X\subset \mathbb{P}^{2}$ of even degree $d\geqslant 2$ there exists a real line $L\subset \mathbb{P}^{2}$ such $X\cap L$ has no real points. We show that the answer is yes if $d=2$ or 4 and no if $n\geqslant 6$.

Let C / k ‾ be a smooth plane curve defined over k ‾ , a fixed algebraic closure of a perfect field k. We call a subfield k ′ ⊆ k ‾ a plane model-field of definition for C if C descends to k ′ as a smooth plane curve over k ′ , that is if there exists a smooth curve C ′ / k ′ defined over k ′ which is k ′ -isomorphic to a non-singular plane model F ( X , Y , Z ) = 0 with coefficients in k ′ , and such that C ′ ⊗ k ′ k ‾ and C are isomorphic. In this paper, we provide (explicit) families of smooth plane curves for which the three fields types; the field of moduli, fields of definition, and plane-models fields of definition are pairwise different.

Let $D$ be a very general curve of degree $d=2\ell-\epsilon$ in $\mathbb{P}^2$, with $\epsilon\in \{0,1\}$. Let $\Gamma \subset \mathbb{P}^2$ be an integral curve of geometric genus $g$ and degree $m$, $\Gamma \neq D$, and let $\nu: C\to \Gamma$ be the normalization. Let $\delta$ be the degree of the \emph{reduction modulo 2} of the divisor $\nu^*(D)$ of $C$. In this paper we prove the inequality $4g+\delta\geqslant m(d-8+2\epsilon)+5$. We compare this with similar inequalities due to Geng Xu and Xi Chen. Besides, we provide a brief account on genera of subvarieties in projective hypersurfaces., Comment: to appear in Cont. Math. ("Selim Krein Centennial"), pp. 1-19. Collaboration has benefitted by "MIUR Excellence Department Project" Dep. of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006, grant 346300 for "IMPAN" from the "Simons Foundation"(code: BCSim-2018-s09), funds "Mission Sustainability 2017 - Fam Curves" CUP E81-18000100005 (Tor Vergata)

Let $(X,H)$ be a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb Z H$, and let $C\in |H|$ be a smooth curve of genus $g$. We give an upper bound on the dimension of global sections of a semistable vector bundle on $C$. This allows us to compute the higher rank Clifford indices of $C$ with high genus. In particular, when $g\geq r^2\geq 4$, the rank $r$ Clifford index of $C$ can be computed by the restriction of Lazarsfeld-Mukai bundles on $X$ corresponding to line bundles on the curve $C$. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank $r$ Clifford index of a degree $d(\geq 5)$ smooth plane curve is $d-4$, which is the same as the Clifford index of the curve., 25 pages, 6 figures, comments are very welcome!

We formulate a theory of shape valid for objects of arbitrary dimension whose contours are path connected. We apply this theory to the design and modeling of viable trajectories of complex dynamical systems. Infinite families of qualitatively similar shapes are constructed giving as input a finite ordered set of characteristic points (landmarks) and the value of a continuous parameter $\kappa \in (0,\infty)$. We prove that all shapes belonging to the same family are located within the convex hull of the landmarks. The theory is constructive in the sense that it provides a systematic means to build a mathematical model for any shape taken from the physical world. We illustrate this with a variety of examples: (chaotic) time series, plane curves, space filling curves, knots and strange attractors., Comment: 19 pages, 10 figures. Discussion on a convexity property added. Accepted to Chaos. Sol. Fract

20 pages, 6 figures, comments welcome; International audience; We study Deraux's non arithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient $X$ of a particular Abelian surface $A$. Using the fact that $A$ is the Jacobian of the Bolza genus $2$ curve, we identify $X$ as the weighted projective plane $\mathbb{P}(1,3,8)$. We compute the equation of the mirror $M$ of the orbifold ball quotient $(X,M)$ and by taking the quotient by an involution, we obtain an orbifold ball quotient surface with mirror birational to an interesting configuration of plane curves of degrees $1,2$ and $3$. We also exhibit an arrangement of four conics in the plane which provides the above-mentioned ball quotient orbifold surfaces.

The main objective of this work is to find the rational points of a plane curve given by the second-degree equation. We illustrate that our method based on affine transformation simplifies the way of finding solutions to complicated second-degree equations.

In this paper, we obtain the L p -boundedness of the Hilbert transform H γ for 1 p ∞ along a variable plane curve ( t , u ( x 1 , x 2 ) γ ( t ) ) , where u is a Lipschitz function with small Lipschitz norm, and γ is a general curve satisfying some suitable smoothness and curvature conditions.

The first goal of this article is to provide a statement of the conditions for geometric continuity of order k, referred in the bibliography as beta-constraints, in terms of Riordan matrices. The second one is to see this new formulation in action to solve a theoretical question about uniqueness of analytic solution for a general and classical problem in plane geometry: the F-chordal problem.

With the development of the advanced precision manufacturing field, the curve profile is becoming more and more complex, and the profile curve evaluation is no longer limited to the two-dimensional plane. Aiming at the difficulty and low efficiency of complex contour evaluation of high-precision parts, this paper proposes a global evaluation method of three-dimensional curve contour. After the curve contour is extracted, dimensioned and calibrated, according to the definition of GB/T1182-2008 plane curve contour, an envelope circle equation is established to evaluate the contour of the complex line. The curve contour theoretical points are used to establish the envelope detection area function, and the position relationship of the measuring points relative to the theoretical contour is evaluated according to the circle function of the envelope group, and the line contour error is determined. Experiments show that the evaluation result is intuitive and can be traced back to the specific area of the curve contour. This method realizes the evaluation of the whole area of the spatially complex curve contour. The method is accurate and efficient, and has overall and unitary characteristics.

Ultrasonic wavefield imaging (UWI) provides insightful spatial information about ultrasonic wave propagation in planar (2-D) space for nondestructive evaluation and structural health monitoring (NDE-SHM) applications. In all materials, the wavefronts of the incident and reflected waves propagate with unique patterns that may be represented by parametrized polar curves in 2-D geometric space. In this paper, a spatial ultrasonic wavefront characterization method based on a parametric curve laser scan is proposed to characterize the spatial ultrasonic wavefront for both isotropic and anisotropic materials. Three parametric curves (circular, hyperbolic, and cyclic-harmonic curves) were considered. Two wavefront characterization process were carried out, namely (i) deciding the parametric equation of the closed-form geometric plane curve via UWI, and (ii) measuring and updating the ultrasound via laser ultrasonic interrogation system (LUIS) and quantifying the values(s) of the predicted parametric curve equation using a temporal cross-correlation technique. The proposed method was tested on pristine aluminum and cross-ply CFRP plates to characterize the spatial incident and reflected wavefronts of the plates. The non-fiber direction region (105°⩽ϕS⩽165°) and the fiber direction region (165°⩽ϕS⩽195°) of the cross-ply CFRP plate were considered in the test. The laser circle scan and the laser cyclic-harmonic curve scan showed the ability to characterize the incident wavefronts of the S0 and A0 modes in the aluminum plate and the CFRP plate, respectively, followed by the laser hyperbolic curve scan. With the promising results obtained in the proposed method, the integration of the parametric curve scanning method into LUIS may provide a new approach to damage detection and useful information for ultrasonic algorithm design in NDE-SHM applications.

In this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study $3$-syzygy arrangements and we present examples that admit unexpected curves., Comment: 19 pages, one figure, version after referee's remarks

The paper reviews recent developments in the study of Alexander invariants of quasi-projective manifolds using methods of singularity theory. Several results in topology of the complements to singular plane curves and hypersurfaces in projective space extended to the case of curves on simply connected smooth projective surfaces.

Let $F\in \mathbb{K}[X, Y ]$ be a polynomial of total degree $D$ defined over a perfect field $\mathbb{K}$ of characteristic zero or greater than $D$. Assuming $F$ separable with respect to $Y$ , we provide an algorithm that computes the singular parts of all Puiseux series of $F$ above $X = 0$ in less than $\tilde{\mathcal{O}}(D\delta)$ operations in $\mathbb{K}$, where $\delta$ is the valuation of the resultant of $F$ and its partial derivative with respect to $Y$. To this aim, we use a divide and conquer strategy and replace univariate factorization by dynamic evaluation. As a first main corollary, we compute the irreducible factors of $F$ in $\mathbb{K}[[X]][Y ]$ up to an arbitrary precision $X^N$ with $\tilde{\mathcal{O}}(D(\delta + N ))$ arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by $F$ with $\tilde{\mathcal{O}}(D^3)$ arithmetic operations and, if $\mathbb{K} = \mathbb{Q}$, with $\tilde{\mathcal{O}}((h+1)D^3)$ bit operations using a probabilistic algorithm, where $h$ is the logarithmic heigth of $F$., Comment: 27 pages, 2 figures

In 1973, Oscar Zariski gave a course at the Ecole Polytechnique (cf. the lecture notes [23] or the monograph [24]), where he discussed in the local case the analogous problem to the construction of the moduli space of algebraic curves of genus g; that is, he proposed to search for moduli spaces with respect to analytic equivalence for germs of irreducible complex analytic plane curves having the same topological type. Zariski recognized that this problem was at that stage very difficult and concentrated his efforts on explicit calculations in some particular cases. Since then, the subject has substantially advanced and our purpose here is to further detail the solution of the moduli problem for plane branches given in [12], using the same framework as Zariski’s, adding to his methods singularity tools that were starting to blossom at this time. The results we present improve several ones found in the literature and shed light over many questions asked by Zariski, using techniques that may be useful to solve other relevant related questions. The exposition is kept as elementary as possible to highlight the beauty and simplicity of the solution of Zariski’s problem and since it is not intended to be a compendium on the subject, several results from algebra and singularity theory will be invoked, giving precise statements and references, where they may be found, without concern about quoting primary sources.

The chapter is devoted to the study of plane curves from the point of view of differential geometry. We focus our attention to classical curves and also on conics, and provide examples and other applications in architectural elements of the mathematical theory.

For calculating the thin-walled closed curved box girder caused by the temperature gradient of the internal force and displacement, based on the fundamental differential equation of the curve beam and the principle of minimum energy, set a reverse statically indeterminate simply supported curve beam as the basic structure, consider the warping effect of the closed curve box girder, and put forward a kind of plane curve beam temperature deformation simple analytical calculation method. Compared with the finite element calculation results, the relative error of the analytical calculation results is less than 5%. It is concluded that the analytical method has sufficient accuracy in calculating the out-of-plane deformation of the thin-walled closed curved box girder under the temperature gradient.

In this paper we present a review of the basic ideas and results concerning asymptotic lines of plane curves. We discuss their different definitions, namely that of a limiting position of tangent lines, of the tangent line at infinity, and finally the one that requires that the distance between points of a curve and asymptotic line tends to 0 as the point moves along an infinite branch of the curve. We also recall the method of determining asymptotes of algebraic curves from the leading coefficients in their equation and provide examples., U ovom radu dajemo pregled osnovnih ideja i rezultata vezanih uz asimptote ravninskih krivulja. Raspravljamo o njihovim različitim definicijama, naime, o definiciji kao o graničnom položaju tangenata, o definiciji kao o tangenti u beskonačnosti, te konačno o definiciji koja zahtijeva da udaljenost između točke krivulje i asimptote teži 0 kako se točka kreće duž beskonačne grane krivulje. Također se prisjećamo metode određivanja asimptota algebarskih krivulja iz vodećih koeficijenata u njihovoj jednadžbi te navodimo primjere.

For a positive integer k and a linearized polynomial L(X), polynomials of the form $P(X)=G(X)^{k}-L(X) \in {\mathbb F}_{q^{n}}[X]$ are investigated. It is shown that when L has a non-trivial kernel and G is a permutation of $\mathbb {F}_{q^{n}}$ , then P(X) cannot be a permutation if $\gcd (k,q^{n}-1)>1$ . Further, necessary conditions for P(X) to be a permutation of $\mathbb {F}_{q^{n}}$ are given for the case that G(X) is an arbitrary linearized polynomial. The method uses plane curves, which are obtained via the multiplicative and the additive structure of $\mathbb {F}_{q^{n}}$ , and their number of rational affine points.

We prove short time existence for higher order curvature flows of plane curves with and without generalised Neumann boundary condition.

International audience; Isolating the singularities of a plane curve is the first step towards computing its topology. For this, numerical methods are efficient but not certified in general. We are interested in developing certified numerical algorithms for isolating the singularities. In order to do so, we restrict our attention to the special case of plane curves that are projections of smooth curves in higher dimensions. This type of curve appears naturally in robotics applications and scientific visualization. In this setting, we show that the singularities can be encoded by a regular square system whose solutions can be isolated with certified numerical methods. Our analysis is conditioned by assumptions that we prove to be generic using transversality theory. We also provide a semi-algorithm to check their validity. Finally, we present experiments, some of which are not reachable by other methods, and discuss the efficiency of our method.

Let $d\in\mathbb{Z}^+t$, $\mathbb{K}$ be a field of characteristic zero and $A$ be a nonempty finite subset of $\mathbb{K}^2$. Denote by $\mathcal{C}_{d,\mathbb{K}}$ the family of algebraic curves of degree $d$ in $\mathbb{K}^2$ and $\mathcal{C}_{\leq d,\mathbb{K}}:=\bigcup_{e=1}^d\mathcal{C}_{e,\mathbb{K}}$. For any $C_1\in \mathcal{C}_{d,\mathbb{K}}$, we say that $C_1$ is determined by $A$ if for any $C_2\in\mathcal{C}{d,\mathbb{K}}$ such that $C_2\cap A\supseteq C_1\cap A$, we have that $C_1=C_2$; we denote by $\mathcal{D}_{d,\mathbb{K}}(A)$ the family of elements of $\mathcal{C}_{d,\mathbb{K}}$ determined by $A$. Beck's theorem establishes that if $\mathbb{K}=\mathbb{R}$ and $A$ is not collinear, then $$|\mathcal{D}_{1,\mathbb{R}}(A)|=\Theta\left(|A|\min_{C\in \mathcal{C}_{1,\mathbb{R}}}|A\setminus C|\right).$$ In this paper we generalize Beck's theorem showing that for all $d\in\mathbb{Z}^+$, there exists a constant $c=c(d)>0$ such that if $\min_{C\in\mathcal{C}_{\leq d,\mathbb{K}}}|A\setminus C|>c,$ then $$|\mathcal{D}_{d,\mathbb{K}}(A)|=\Theta_d\left(|A|^d\prod_{e=1}^d\left(\min_{C\in \mathcal{C}_{\leq e,\mathbb{K}}}|A\setminus C|\right)^{d-e+1}\right).$$

Let $K$ be the function field of a curve over a finite field of odd characteristic. We investigate using $L$-functions of Galois extensions of $K$ to effectively recover $K$. When $K$ is the function field of the projective line with four rational points removed, we show how to use $L$-functions of a ray class field to effectively recover the removed points up to automorphisms of the projective line. When $K$ is the function field of a plane curve, we show how to effectively recover the equation of that curve using $L$-functions of Artin-Schreier extensions of $K$., Comment: 23 pages; comments welcome

Based on the assumption that the contours of the tested workpiece are all plane curves composed of straight lines and circular arcs, this article first adopts the least square method to establish a mathematical model for segmented fitting, and calculate the workpiece by using Matlab program to obtain the parameter values of the contour line detected in the horizontal state. Furthermore, due to the deviation of the angle and position of the workpiece measurement placement, the difference in the parameter calculation values will be brought about. At this time, the inclination angle during the measurement needs to be calculated first, and a model is established according to the tilt angle to horizontally correct the measurement data, and then fit, compare, and modify according to the corrected data, and finally obtain the parameter values of the contour line.

In this paper, we introduce the notions of equiaffine arclength and curvature with fractional order for a plane curve and compare them with the standard ones. In terms of the equiaffine curvature with fractional order we obtain an equiaffine Frenet formula and then construct an analogue of the Fundamental Theorem. The plane curves of constant equiaffine curvature with fractional-order are classified. Several examples are also illustrated.

In this paper, we introduce a new class of curves that we call helical extension curve in Euclidean space. Then we investigate the differential geometric properties of the helical extension curves. The main result of the paper is that the helical extension curve of a helix is a plane curve. Moreover, we give several simple characterizations of helical extension curve of special curves such as spherical curve.

We show that if a homogeneous polynomial $f$ in $n$ variables has Waring rank $n+1$, then the corresponding projective hypersurface $f=0$ has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of $f$. We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5., v4: We have added Lemma 2.2, which is necessary for a clear proof of the unicity of the integer $k$ in Theorem 2.4. We also attribute Proposition 4.1 to Neriman Tokcan, since we were informed that it appears as Theorem 3.1 in her paper 'On the Waring rank of binary forms', arXiv:1610.09065

A criterion for the existence of a plane model with two non-smooth Galois points for algebraic curves is presented, which is a generalization of Fukasawa's criterion for two smooth Galois points. Owing to this generalized criterion, multiplicities and order sequences at Galois points can be described in detail., 11pages

We study the asymptotic proportion of smooth plane curves over a finite field $\mathbb{F}_q$ which are tangent to every line defined over $\mathbb{F}_q$. This partially answers a question raised by Charles Favre. Our techniques include applications of Poonen's Bertini theorem and Schrijver's theorem on perfect matchings in regular bipartite graphs. Our main theorem implies that a random smooth plane curve over $\mathbb{F}_q$ admits a transverse $\mathbb{F}_q$-line with very high probability., 19 pages

Tanks containing liquid propellants for rocket engines can be considered as shells whose thin walls are surfaces of revolution. Such surfaces result from a plane curve which rotates about some straight line lying in the plane which contains the curve. The structural analysis of tanks is presented here by using Roark’s formulas for stress and strain. Loads due to propellant sloshing and slosh-suppression devices are also discussed.

Let $\mathcal{C}$ be a plane curve given by an equation $f(x,y)=0$ with $f\in K[x][y]$ a monic squarefree polynomial. We study the problem of computing an integral basis of the algebraic function field $K(\mathcal{C})$ and give new complexity bounds for three known algorithms dealing with this problem. For each algorithm, we study its subroutines and, when it is possible, we modify or replace them so as to take advantage of faster primitives. Then, we combine complexity results to derive an overall complexity estimate for each algorithm. In particular, we modify an algorithm due to B\"ohm et al. and achieve a quasi-optimal runtime., Comment: Preliminary version

We study complex projective plane curves with a given group of automorphisms. Let $G$ be a simple primitive subgroup of $PGL(3, \mathbb{C})$, which is isomorphic to $A_{6}$, $A_{5}$ or $PSL(2, \mathbb{F}_{7})$. We obtain a necessary and sufficient condition on $d$ for the existence of a nonsingular projective plane curve of degree $d$ invariant under $G$. We also study an analogous problem on integral curves., 32 pages

$$E^{2}_{1}$$ be the 2-dimensional pseudo-Euclidean space of index 1, $$G=Sim_{L}(E^{2}_{1})$$ be the group of all linear pseudo-similarities of $$E^{2}_{1}$$ and $$G=Sim_{L}^{+}(E^{2}_{1})$$ be the group of all orientation-preserving linear pseudo-similarities of $$E^{2}_{1}$$ . In this paper, groups $$Sim_{GL}^{+}(E^{2}_{1})$$ and $$Sim_{GL}(E^{2}_{1})$$ are defined. For the groups $$G=Sim_{GL}^{+}(E^{2}_{1}),Sim_{GL}(E^{2}_{1})$$ , $$Sim_{L}(E^{2}_{1}),$$ $$Sim_{L}^{+}(E^{2}_{1})$$ , G-invariants of paths in $$E^{2}_{1}$$ are investigated. Using hyperbolic numbers, a method to detect G-similarities of paths and curves is presented. We give an evident form of a path and a curve in terms of their G-invariants. For two paths and two curves with the common G-invariants, evident forms of all linear pseudo-similarity transformations, carrying the paths and the curves, are obtained.

This paper focuses on preliminary work related to the discovery of single degree-of-freedom mechanism paths useful for dynamic locomotion tasks. The objective is to bridge a gap between kinematic specifications and emerging dynamic behaviors. This is accomplished by formulating a set of ordinary differential equations that includes essential mechanism characteristics (path traced, mechanical advantage) but excludes all physical mechanism parameters (topology, link lengths). The dynamics represent a rotation constrained body propelled by a foot that is attached to that body by a user-defined path. The foot is powered by a series-elastic actuator acting through a mechanical advantage function that is defined across the length of the path. Through this framework, a range of user-defined paths were tested for effective locomotion on flat and complex terrains. Foot paths and mechanical advantage functions exist outside of any mechanical design, with the goal to discover paradigms worth instantiating into physical mechanisms, a task reserved for kinematic synthesis. This work would empower existing kinematic synthesis techniques to achieve dynamic requirements. In other words, kinematic requirements are transformed from an end to a means. Their dynamic utility would be evaluated by the framework presented in this paper rather than pursued by themselves.

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 Poincar\'e polynomial of the compactified space., Comment: Comments are welcome