2006 Fiscal Year Final Research Report Summary
Special linear systems on Riemann surfaces
| Project/Area Number |
17540160
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Yamaguchi University |
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Principal Investigator |
KATO Takao Yamaguchi University, Graduate School of Science and Engineering, Professor, 大学院理工学研究科, 教授 (10016157)
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| Co-Investigator(Kenkyū-buntansha) |
MASUMOTO Makoto Yamaguchi University, Graduate School of Science and Engineering, Professor, 大学院理工学研究科, 教授 (50173761)
KURIYAMA Ken Yamaguchi University, Graduate School of Science and Engineering, Professor, 大学院理工学研究科, 教授 (10116717)
YANAGIHARA Hiroshi Yamaguchi University, Graduate School of Science and Engineering, Associate Professor, 大学院理工学研究科, 助教授 (30200538)
HOMMA Masaaki Kanagawa University, Faculty of Engineering, Professor, 工学部, 教授 (80145523)
OBUCHI Akira Tokushima University, Faculty of Integrated Arts and Sciences, Professor, 総合科学部, 教授 (10211111)
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| Project Period (FY) |
2005 – 2006
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| Keywords | compact Riemann surfaces / algebraic curves / meromorphic functions / linear systems / gonality / error-correcting coding theory |
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| Research Abstract |
We study classification problems for compact Riemann surfaces through the existence of meromorphic functions on them and conformal invarinats.
1. Let C be a compact Riemann surface of genus g. The minimal degree of pencils on is said to be the gonality of C and denoted by gon(C). This quantity is a conformal invarinat. It is well-known that 2 【less than or equal】 gon(C) 【less than or equal】 [(g + 3)/2]. On the other hand, let s_2(C) be the minimal degree of simple nets on C. While s_2(C) satisfies (3 + √<8g+1>)/2 【less than or equal】s_2(C) 【less than or equal】 g + 2, these two quantities relate strongly. As a matter of fact, If gon(C) = 2, then s_2(C) = g + 2 and vice versa. Furthermore, in case g 【greater than or equal】 6, that C is elliptic-hyperelliptic if and only if s_2(C) = g + 1. In this project, we show that for almost all g, there is no C such that s_2(C) = g. Moreover, in the case where C is 4-gonal of genus 9 with the scrollar invariant (4,1,1), we decided s_2(C). This case seems to be the most complicated case among 4-gonal cases.
2. Let F_q be a finite fields with q elements and C ⊂F^n_q be a linear [n,k,d]q code. Let n_q(k,d) be the minimum of the code lengths for fixed k, d. There is an upper bound of n_q(k,d) known as the Griesmer bound. In this project, we show that for some range of d's, n_q(k,d) (for k = 5,6) is equal to the Griesmer bound minus 1.
As a generalization of the notion of the Weierstrass point, one can define the notion of Weierstrass n-tuple by choosing appropriate n points. Using the pure gaps of Weierstrass n-tuple, we obtained an estimate of the minimal distance of the Goppa codes.
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Research Products
(12 results)
