Near-MDS codes arising from algebraic curves

Every elliptic quartic Γ₄ of PG(3,q) with nGF(q)-rational points provides a near-MDS code C of length n and dimension 4 such that the collineation group of Γ₄ is isomorphic to the automorphism group of C. In this paper we assume that GF(q) has characteristic p>3. We classify the linear collineation groups of PG(3,q) which can preserve an elliptic quartic of PG(3,q). Also, we prove for q?113 that if the j-invariant of Γ₄ does not disappear, then C cannot be extended in a natural way by adding a point of PG(3,q) to Γ₄.     

Low-degree points on Hurwitz-Klein curves

We investigate low-degree points on the Fermat curve of degree 13, the Snyder quintic curve and the Klein quartic curve. We compute all quadratic points on these curves and use Coleman's effective Chabauty method to obtain bounds for the number of cubic points on each of the former two curves.

     

Fixed points of automorphisms of real algebraic curves

We bound the number of fixed points of an automorphism of a real curve in terms of the genus and the number of connected components of the real part of the curve. Using this bound, we derive some consequences concerning the maximum order of an automorphism and the maximum order of an abelian group of automorphisms of a real curve. We also bound the full group of automorphisms of a real hyperelliptic curve. Work supported by the European Community’s Human Potential Programme under contract HPRN-CT-2001-00271, RAAG.     

A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals

We give a class of quadratic systems without rational first integral which contains irreducible algebraic solutions of arbitrarily high degree. The construction gives a negative answer to a conjecture of Lins Neto and others.

     

The Automorphism Group of Plane Algebraic Curves with Singer Automorphisms

The main result in Cossidente and Siciliano (J. Number Theory, Vol. 99 (2003) pp. 373–382) states that if a Singer subgroup of PGL(3,q) is an automorphism group of a projective, geometric irreducible, non-singular plane algebraic curve then either or . In the former case is projectively equivalent to the curve with equation X^q+1Y+Y^q+1+X=0 studied by Pellikaan. Furthermore, the curve has a very nice property from Finite Geometry point of view: apart from the three distinguished points fixed by the Singer subgroup, the set of its -rational points can be partitioned into finite projective planes . In this paper, the full automorphism group of such curves is determined. It turns out that is the normalizer of a Singer group in .     

Clifford's inequality for real algebraic curves

We improve Clifford's Inequality for real algebraic curves. As an application we improve Harnack's Inequality for real space curves having a certain number of pseudo-lines. Another application involves the number of ovals that a real space curve can have.     

Cubic algebraic spline curves design

In this paper we propose a construction method of the planar cubic algebraic spline curve with endpoint interpolation conditions and a specific analysis of its properties. The piecewise cubic algebraic curve has G ² continuous contact with the control polygon at two endpoints and is G ² continuous between each segments of itself. The process of this method is simple and clear, and provides a new way of thinking to design implicit curves.     

Plane algebraic curves with many cusps,with an appendix by Eugenii Shustin

The maximum number \(k(d)\) of cusps on a plane algebraic curve of degree \(d\) is an open classical problem that dates back to the nineteenth century. A related open problem is the asymptotic value ( \(a.v.\) ) of the number of cusps on plane curves, that is \(a.v.=\limsup _{d\rightarrow \infty } k(d)/{d^2}\) . In this paper, we improve the best known lower bound for the asymptotic value by constructing curves with the largest known number of cusps for infinitely many degrees. Some particular curves of relatively low degree with many cusps are constructed too. The “Appendix” to this paper is devoted to the case of degree 11 and it is due to E. Shustin.     

On theta characteristics of real algebraic curves

Real theta characteristics of a real algebraic curve are studied. The numbers of even and odd real theta characteristics are calculated. These numbers depend on the topological characteristics of the curve only. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 403–106, September, 1998.     

椭圆曲线密码的快速算法

80年代,椭圆曲线理论被引入数据加密领域,形成了一种新的公开密钥体制即椭圆曲线密码体制(ECC).该体制中,最耗时的运算是倍点运算也就是椭圆曲线上的点与一个整数的乘法运算.因此倍点运算的快速计算是椭圆曲线密码快速实现的关键.本文提出一种计算kP新的算法,使效率提高38%以上.     

The Bezout number of piecewise algebraic curves

Based on the discussion of the number of roots of univariate spline and the common zero points of two piecewise algebraic curves, a lower upbound of Bezout number of two piecewise algebraic curves on any given non-obtuse-angled triangulation is found. Bezout number of two piecewise algebraic curves on two different partitions is also discussed in this paper.     

Invariant algebraic curves and rational first integrals of holomorphic foliations in CP(2)

In this paper, using the tools of algebraic geometry we provide sufficient conditions for a holomor-phic foliation in CP(2) to have a rational first integral. Moreover, we obtain an upper bound of the degrees of invariant algebraic curves of a holomorphic foliation in CP(2). Then we use these results to prove that any holomorphic foliation of degree 2 does not have cubic limit cycles.     

On the Halphen transform of algebraic space curves

The Halphen transform of a plane curve is the curve obtained by intersecting the tangent lines of the curve with the corresponding polar lines with respect to some conic. This transform was introduced by Halphen as a branch desingularization method in [5 Halphen, G. H. (1876). Sur une série de courbes analogues aux développées. J. Math. Pures Appl. 3e série, tome 2:87–144. [Google Scholar]] and has also been studied in [2 Coolidge, J. L. (2004). A Treatise on Algebraic Plane Curves. New York: Dover Publications, Inc., 1959, xxiv+513 pp. [Google Scholar], 8 Josse, A. (1995). Transformation d’Halphen. (French) [The Halphen transform]. Commun. Algebra 23(12):4343–4364.[Taylor & Francis Online] , [Google Scholar]]. We extend this notion to the Halphen transform of a space curve and study several of its properties (birationality, degree, rank, class, desingularization).     

Galois points for double-Frobenius nonclassical curves

We determine the distribution of Galois points for plane curves over a finite field of q elements, which are Frobenius nonclassical for different powers of q. This family is an important class of plane curves with many remarkable properties. It contains the Dickson–Guralnick–Zieve curve, which has been recently studied by Giulietti, Korchmáros, and Timpanella from several points of view. A problem posed by the second author in the theory of Galois points is modified.     

Proper partial geometries with Singer groups and pseudogeometric partial difference sets

A partial geometry admitting a Singer group G is equivalent to a partial difference set in G admitting a certain decomposition into cosets of line stabilizers. We develop methods for the classification of these objects, in particular, for the case of abelian Singer groups. As an application, we show that a proper partial geometry Π=pg(s+1,t+1,2) with an abelian Singer group G can only exist if t=2(s+2) and G is an elementary abelian 3-group of order ³(s+1) or Π is the Van Lint-Schrijver partial geometry. As part of the proof, we show that the Diophantine equation (m³−1)/2=(²rw−1)/(r²−1) has no solutions in integers m,r?1, w?2, settling a case of Goormaghtigh's equation.     

一类平面系统的定性分析

利用平面系统的定性理论 ,提供系统的全局分析 ,讨论了一类平面系统奇点的稳定性和极限环问题 ,给出奇点稳定和极限环唯一的充分条件