arXiv math/04110972004-11-04math.AG

Real plane algebraic curves with asymptotically maximal number of even ovals

Erwan brugalle

备注：: 12 pages, 10 figures

英文摘要：

It is known for a long time that a nonsingular real algebraic curve of degree 2k in the projective plane cannot have more than 7/2*k^2-9/4*k+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/k^2 tends to 7/4$ as k tends to infinity, where p is the number of even ovals of the curves. We also show that the same kind of result is valid dealing with odd ovals.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

On maximally inflected hyperbolic curves

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers $d$ and $r$ such that $4\leq r \leq 2d^2-2d$, there is a non-singular hyperbolic curve of degree $2d$ in $\mathbb R^2$ with exactly $r$ line segments in the boundary of its convex hull. We also give a complete classification of possible distributions of inflection points among the ovals of a maximally inflected non-singular hyperbolic curve of degree $6$.

2014-05-13math.AGmath.GT

A tropical construction of a family of real reducible curves

We give a constructive proof using tropical modifications of the existence of a family of real algebraic plane curves with asymptotically maximal numbers of even ovals.

2015-10-12math.AG

来自平面图的极大Mumford曲线

Maximal Mumford Curves from Planar Graphs

本文研究极大Mumford曲线的构造，通过在典范曲线的Hilbert概型中利用平面图曲线的一阶形变，构建了具有全维数的MM曲线族。

2024-04-18math.AGmath.CO

Generic pointed quartic curves in $\mathbb{R}\mathbb{P}^{2}$ and uninodal dessins

In this article we obtain a rigid isotopy classification of generic pointed quartic curves $(A,p)$ in $\mathbb{R}\mathbb{P}^{2}$ by studying the combinatorial properties of dessins. The dessins are real versions, proposed by S. Orevkov, of Grothendieck's dessins d'enfants. This classification contains 20 classes determined by the number of ovals of $A$, the parity of the oval containing the marked point $p$, the number of ovals that the tangent line $T_p A$ intersects, the nature of connected components of $A\setminus T_p A$ adjacent to $p$, and in the maximal case, on the convexity of the pos…

2018-04-13math.AG

Asymptotically maximal families of hypersurfaces in toric varieties

A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with $\mathbb{Z}_2$ coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope $Δ$ there are families of hypersurfaces with the Newton polytopes $(λΔ)_{λ\in \mathbb{N}}$ that are asymptotically maximal when $λ$ tends to infinity. We also show that these results ge…

2005-04-23math.AG

On the number of components of a complete intersection of real quadrics

Our main results concern complete intersections of three real quadrics. We prove that the maximal number $B^0_2(N)$ of connected components that a regular complete intersection of three real quadrics in $\Bbb{P}^N$ can have differs at most by one from the maximal number of ovals of the submaximal depth $[(N-1)/2]$ of a real plane projective curve of degree $d=N+1$. As a consequence, we obtain a lower bound \smash{$\frac14 N^2+O(N)$} and an upper bound \smash{$\frac38 N^2+O(N)$} for $B^0_2(N)$.

2012-01-14math.AG

更多相关论文收起更多论文

Determinantal representations of hyperbolic curves via polynomial homotopy continuation

A smooth curve in the real projective plane is hyperbolic if its ovals are maximally nested. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to compute such representations numerically. Our method works by lifting paths from the space of hyperbolic polynomials to a branched cover in the space of pairs of symmetric matrices.

2016-07-04math.AGmath.OC

A note on Harris Morrison sweeping families of maximal gonality

Harris and Morrison constructed semistable families f:F \to Y of k-gonal curves of genus g such that for every k the corresponding modular curves give a sweeping family in the k-gonal locus in the moduli space. Their construction depends on the choice of a smooth curve X. We show that if the genus g(X) is sufficiently high with respect to g, then the ratio K_F^2 / χ(O_F) is 8 asymptotically with respect to g(X). We show also that if the gonality is maximal and some conjectured estimates of Harris and Morrison hold, the slope of the fibration f: F\to Y is 12 asymptotically with respect to g and…

2013-03-11math.AG

$\mathbb{RP}^4$ 中极大实代数超曲面的四个族

Four families of maximal real algebraic hypersurfaces in $\mathbb{RP}^4$

本文用 Viro 组合拼接法构造 $\mathbb{RP}^4$ 中四个偶次极大实代数超曲面族，并证明其相关二重覆盖的实部欧拉示性数与复部符号差不必相等，甚至可渐近不同。

2023-10-27math.AG

用渐近大 Betti 数拼凑实代数超曲面

Patchworking real algebraic hypersurfaces with asymptotically large Betti numbers

本文基于 Viro 的 Patchwork 提出递归构造实射影代数超曲面族的方法，证明其 Betti 数可渐近超过相应 Hodge 数，并构造出 Betti 数在次数和环境维数上均极大的族。

2022-03-23math.AG

On the complexity group of stable curves

In this paper, we study combinatorial properties of stable curves. To the dual graph of any nodal curve, it is naturally associated a group, which is the group of components of the Néron model of the generalized Jacobian of the curve. We study the order of this group, called the complexity. In particular, we provide a partial characterization of the stable curves having maximal complexity, and we provide an upper bound, depending only on the genus $g$ of the curve, on the maximal complexity of stable curves; this bound is asymptotically sharp for $g\gg 0$. Eventually, we state some conjectures…

2008-08-11math.AGmath.CO

Tower of algebraic function fields with maximal Hasse-Witt invariant and tensor rank of multiplication in any extension of $\mathbb{F}_2$ and $\mathbb{F}_3$

Up until now, it was recognized that a large number of 2-torsion points was a technical barrier to improve the bounds for the symmetric tensor rank of multiplication in every extension of any finite field. In this paper, we show that there are two exceptional cases, namely the extensions of $\mathbb{F}_2$ and $\mathbb{F}_3$. In particular, using the definition field descent on the field with 2 or 3 elements of a Garcia-Stichtenoth tower of algebraic function fields which is asymptotically optimal in the sense of Drinfel'd-Vladut and has maximal Hasse-Witt invariant, we obtain a significant imp…

2014-09-11cs.ITmath.AGmath.IT