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Satoru Fukasawa - One of the best experts on this subject based on the ideXlab platform.
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galois points for a Plane Curve in characteristic two
Journal of Pure and Applied Algebra, 2014Co-Authors: Satoru FukasawaAbstract:Abstract Let C be an irreducible Plane Curve. A point P in the projective Plane is said to be Galois with respect to C if the function field extension induced by the projection from P is Galois. We denote by δ ′ ( C ) the number of Galois points contained in P 2 ∖ C . In this article we will present two results with respect to determination of δ ′ ( C ) in characteristic two. First we determine δ ′ ( C ) for smooth Plane Curves of degree a power of two. In particular, we give a new characterization of the Klein quartic in terms of δ ′ ( C ) . Second we determine δ ′ ( C ) for a generalization of the Klein quartic, which is related to an example of Artin–Schreier Curves whose automorphism group exceeds the Hurwitz bound. This Curve has many Galois points.
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Galois points for a non-reflexive Plane Curve of low degree
Finite Fields and Their Applications, 2013Co-Authors: Satoru FukasawaAbstract:Abstract We consider a Plane Curve C. A point P in the projective Plane is said to be Galois with respect to C if the projection from the point P induces a Galois extension of function fields. In this article, we give a new example of a Plane Curve C of degree q + 1 such that the set of Galois points for C coincides with the one of F q -rational points of P 2 . This Curve appears in the classification list of ‘non-reflexive Plane Curves of low degree’ in positive characteristic. We also determine the sets of Galois points for such low-degree Plane Curves.
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complete determination of the number of galois points for a smooth Plane Curve
arXiv: Algebraic Geometry, 2010Co-Authors: Satoru FukasawaAbstract:Let $C$ be a smooth Plane Curve. A point $P$ in the projective Plane is said to be Galois with respect to $C$ if the function field extension induced from the point projection from $P$ is Galois. We denote by $\delta(C)$ (resp. $\delta'(C)$) the number of Galois points contained in $C$ (resp. in $\mathbb P^2 \setminus C$). In this article, we determine the numbers $\delta(C)$ and $\delta'(C)$ in any remaining open cases. Summarizing results obtained by now, we will have a complete classification theorem of smooth Plane Curves by the number $\delta(C)$ or $\delta'(C)$. In particular, we give new characterizations of Fermat Curve and Klein quartic Curve by the number $\delta'(C)$.
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Singular Plane Curves with infinitely many Galois points
Journal of Algebra, 2010Co-Authors: Satoru Fukasawa, Takehiro HasegawaAbstract:For a Plane Curve C, we call a point P∈P2 a Galois point with respect to C if the point projection from P induces a Galois extension of function fields. We give an example of a singular Plane Curve having infinitely many inner and outer Galois points. We also classify Plane Curves whose general points are inner Galois points. Before our results, known examples in the theory of Galois points have only finitely many Galois points, except trivial cases.
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Galois points for a Plane Curve in arbitrary characteristic
Geometriae Dedicata, 2008Co-Authors: Satoru FukasawaAbstract:In 1996, Hisao Yoshihara introduced a new notion in algebraic geometry: a Galois point for a Plane Curve is a point from which the projection induces a Galois extension of function fields. Yoshihara has established various new approaches to algebraic geometry by using Galois point or generalized notions of it. It is an interesting problem to determine the distribution of Galois points for a given Plane Curve. In this paper, we survey recent results related to this problem.
Yoshihiro Ônishi - One of the best experts on this subject based on the ideXlab platform.
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Arithmetical Power Series Expansion of the Sigma Function for a Plane Curve
Proceedings of the Edinburgh Mathematical Society, 2018Co-Authors: Yoshihiro ÔnishiAbstract:AbstractThe Weierstrass function σ(u) associated with an elliptic Curve can be generalized in a natural way to an entire function associated with a higher genus algebraic Curve. This generalized multivariate sigma function has been investigated since the pioneering work of Felix Klein. The present paper shows Hurwitz integrality of the coefficients of the power series expansion around the origin of the higher genus sigma function associated with a certain Plane Curve, which is called an (n, s)-Curve or a Plane telescopic Curve. For the prime (2), the expansion of the sigma function is not Hurwitz integral, but its square is. This paper clarifies the precise structure of this phenomenon. In Appendix A, computational examples for the trigonal genus 3 Curve ((3, 4)-Curve) y3 + (μ1x + μ4)y2 + (μ2x2 + μ5x + μ8)y = x4 + μ3x3 + μ6x2 + μ9x + μ12 (where μj are constants) are given.
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hurwitz integrality of power series expansion of the sigma function for a Plane Curve
arXiv: Number Theory, 2015Co-Authors: Yoshihiro ÔnishiAbstract:This paper shows Hurwitz integrality of the coefficients of expansion at the origin of the sigma function \(\sigma(u)\) associated to a certain Plane Curve which should be called a Plane telescopic Curve. For the prime \(2\), the expansion of \(\sigma(u)\) is not Hurwitz integral, but \(\sigma(u)^2\) is. This paper clarifies the precise structure of this phenomenon. Throughout the paper, computational examples for the trigonal genus three Curve (\((3,4)\)-Curve) \(y^3+(\mu_1x+\mu_4)y^2+(\mu_2x^2+\mu_5x+\mu_8)y=x^4+\mu_3x^3+\mu_6x^2+\mu_9x+\mu_{12}\) (\(\mu_j\) are constants) are given.
Arkadiusz Ploski - One of the best experts on this subject based on the ideXlab platform.
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a bound for the milnor number of Plane Curve singularities
Open Mathematics, 2014Co-Authors: Arkadiusz PloskiAbstract:Let f = 0 be a Plane algebraic Curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the Curves f = 0 for which μ0(f) = (d−1)2 − [d/2].
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a bound for the milnor number of Plane Curve singularities
arXiv: Algebraic Geometry, 2013Co-Authors: Arkadiusz PloskiAbstract:Let $f=0$ be a Plane algebraic Curve of degree $d>1$ with an isolated singular point at the origin of the complex Plane. We show that the Milnor number $\mu_0(f)$ is less than or equal to $(d-1)^2-\left[\frac{d}{2}\right]$, unless $f=0$ is a set of $d$ concurrent lines passing through 0. Then we characterize the Curves $f=0$ for which $\mu_0(f)=(d-1)^2-\left[\frac{d}{2}\right]$.
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invariants of Plane Curve singularities and newton diagrams
Universitatis Iagellonicae Acta Mathematica, 2012Co-Authors: Pierrette Cassounogues, Arkadiusz PloskiAbstract:We present an intersection-theoretical approach to the invariants of Plane Curve singularities μ, δ, r related by the Milnor formula 2δ = μ + r − 1. Using Newton transformations we give formulae for μ, δ, r which imply planar versions of well-known theorems on nondegenerate singularities.
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polar invariants of Plane Curve singularities intersection theoretical approach
Demonstratio Mathematica, 2010Co-Authors: Janusz Gwoździewicz, Andrzej Lenarcik, Arkadiusz PloskiAbstract:This article, based on the talk given by one of the authors at the Pierret- tefest in Castro Urdiales in June 2008, is an overview of a number of recent results on the polar invariants of Plane Curve singularities.
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characterization of non degenerate Plane Curve singularities
arXiv: Algebraic Geometry, 2007Co-Authors: Evelia Garcia R Barroso, Andrzej Lenarcik, Arkadiusz PloskiAbstract:We characterize Plane Curve germes non-degenerate in Kouchnirenko's sense in terms of characteristics and intersection multiplicities of branches.
Allen Tannenbaum - One of the best experts on this subject based on the ideXlab platform.
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on affine Plane Curve evolution
Journal of Functional Analysis, 1994Co-Authors: Guillermo Sapiro, Allen TannenbaumAbstract:Abstract An affine invariant Curve evolution process is presented in this work. The evolution studied is the affine analogue of the Euclidean Curve Shortening flow. Evolution equations, for both affine and Euclidean invariants, are developed. An affine version of the classical (Euclidean) isoperimetric inequality is proved. This inequality is used to show that in the case of affine evolution of convex Plane Curves, the affine isoperimetric ratio is a non-decreasing function of time. Convergence of this affine isoperimetric ratio to the ellipse′s value (8π 2 ), as well as convergence, in the Hausdorff metric, of the evolving Curve to an ellipse, is also proved.
Lionel Moisan - One of the best experts on this subject based on the ideXlab platform.
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affine Plane Curve evolution a fully consistent scheme
IEEE Transactions on Image Processing, 1998Co-Authors: Lionel MoisanAbstract:We present an accurate numerical scheme for the affine Plane Curve evolution and its morphological extension to grey-level images. This scheme is based on the iteration of a nonlocal, fully affine invariant and numerically stable operator, which can be exactly computed on polygons. The properties of this operator ensure that a few iterations are sufficient to achieve a very good accuracy, unlike classical finite difference schemes that generally require a lot of iterations. Convergence results are provided, as well as theoretical examples and experiments.
