2011 Fiscal Year Final Research Report
A study of algebraic curves from viewpoints of the coding theory and the finite geometry
| Project/Area Number |
21540051
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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 |
Algebra
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| Research Institution | Kanagawa University |
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Principal Investigator |
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| Research Collaborator |
KIM Seonjeong 国立慶尚大学校, 自然科学大学, 教授
CHEON Eunju 国立慶尚大学校, 自然科学大学, ポスドク研究員
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| Project Period (FY) |
2009 – 2011
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| Keywords | 平面代数曲線 / 有限体 / 有理点の個数 / Sziklai予想 |
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| Research Abstract |
We found the following facts. Let Fq be the finite field of q-elements, and F(X, Y, Z) a homogeneous polynomial of degree d over Fq,. We consider the plane curve C defined by F(X, Y, Z)=0.Suppose that C has no Fq-linear component. Let Nq(C) denote the number of the set of Fq. points of C.
Then we have Theorem. Unless C is a curve defined over F4 which is projectively equivalent to(X+Y+Z) 4+(XY+YZ+ZX) 2+XYZ(X+Y+Z), the inequality Nq(C).(d-1) q+1 holds true.
Moreover, there exists a nonsingular curve of degree d which attains the bound above if and only if d is one of the following numbers{q+2, q+1, q, q-1,√<q+1>(if q is a square), 2}.
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