Real plane algebraic curves

We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi-definite nondefinite or definite. We present a discussion about isolated points. By means of the P operator, these points can be easily identified for curves defined by minimal polynomials of order bigger than one. We also discuss the conditions that a curve must satisfy in order to have a minimal polynomial. Finally, we list the most relevant topological properties of affine and projective, complex and real plane algebraic curves.     

Geometry of plane curves

In this paper we obtain, for a class of plane curves, extensions of the well-known relation of inflection points, double points and bitangencies established by Fabricius-Bjerre for closed curves.     

Intermediate links of plane curves

For a smooth complex curve C ? ?² we consider the link L_r = C ∩ ?B_r, where B_r denotes an Euclidean ball of radius r > 0. We prove that the diagram D_r obtained from L_r by a complex stereographic projection satisfies χ(C ∩B_r) = rot(D_r)?wr(D_r). As a consequence we show that if D_r has no negative Seifert circles and L_r is strongly quasipositive and fibered, then the Yamada–Vogel algorithm applied to D_r yields a quasipositive braid.     

On multi-Frobenius non-classical plane curves

In this paper, we characterize the plane curves over \mathbb F_q{\mathbb {F}_q} which are Frobenius non-classical for different powers of q.     

Varieties of strange plane curves

Abstract

Let A be a commutative ring with identity, let X, Y be indeterminates and let F(X,Y), G(X, Y) ∈ A[X, Y] be homogeneous. Then the pair F(X, Y), G(X, Y) is said to be radical preserving with respect to A if Rad((F(x, y), G(x, y))R) = Rad((x,y)R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ?-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ? Q of A such that ht(Q/P) = 2.     

Singularities of plane algebraic curves

We give an exposition of some of the basic results on singularities of plane algebraic curves, in terms of polynomials and formal power series.     

Isogeometric spline reconstruction of plane curves

We establish conditions for the isogeometric reconstruction of plane curves by using parabolic and cubic parametric splines of minimal defect.     

General properties of spiral plane curves

The paper treats plane curves with monotone curvature {spirals). Some familiar facts of the geometry of spirals are generalized by way of removing requirements that the curvature be continuous and have constant sign. Necessary and sufficient conditions for the existence of a spiral with given boundary conditions are considered. The existence of bipolar equation of a spiral is also discussed. Bibliography: 8 titles.     

Ultradifferentiable functions on smooth plane curves

The regularity, in the sense of ultradifferentiability, of real functions of two variables is determined in terms of the regularity of their restrictions to a given family of smooth plane curves. The special case of line segments reduces to the main result in [Proc. Amer. Math. Soc. 127 (1999) 2099-2104]. As a consequence, the Bochnak-Siciak theorem on real analyticity is obtained. A formal analog of one of the results provides a generalization of the two-variable case of the Abhyankar and Moh [J. Reine Angew. Math. 241 (1970) 27-33] theorem.     

A note on plane pointless curves

Let $d(q)$ denote the minimal degree of a smooth projective plane curve that is defined over the finite field ${\mathbb{F}}_q$ and does not contain ${\mathbb{F}}_q$ rational points. We are interested in the asymptotic behavior of $d(q)$ for $q\to \infty$. To the best of the author's knowledge the problem of estimating the asymptotic behavior of $d(q)$ was not considered previously. In this note we establish the following bounds:(1)$\frac{1}{4}⩽\underset{q\to \infty }{\underline{\lim }}{\log }_{q}d(q)⩽\frac{1}{3}.$ More specifically, for every characteristic $p>3$ we construct a sequence of pointless Fermat curves$x^{d_k}+y^{d_k}+z^{d_k}=0,\text{over }{\mathbb{F}}_{p^{m_k}},$ such that ${\lim }_{k\to \infty }{\log }_{p^{m_k}}{d}_k=1/3$.     

Relative node polynomials for plane curves

We generalize the recent work of S.?Fomin and G.?Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and ?? nodes is given by a polynomial in d, provided ?? is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves that, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined ??relative node polynomial?? in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary ??, and use it to present explicit formulas for ????6. We also give a threshold for polynomiality, and compute the first few leading terms for any???.     

On a property of plane curves

Let γ:[0,1]→²[0,1] be a continuous curve such that γ(0)=(0,0), γ(1)=(1,1), and γ(t)∈²(0,1) for all t∈(0,1). We prove that, for each n∈N, there exists a sequence of points Ai, 0?i?n+1, on γ such that A₀=(0,0), An+1=(1,1), and the sequences and , 0?i?n, are positive and the same up to order, where π₁, π₂ are projections on the axes.