2015 Fiscal Year Final Research Report
Algebraic curves over finite fields and their applications to coding theory and finite geometry
| Project/Area Number |
24540056
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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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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| Project Period (FY) |
2012-04-01 – 2016-03-31
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| Keywords | 代数多様体 / 有限体 / 射影空間 / 超曲面 |
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| Outline of Final Research Achievements |
For the past decade, we have had a great interest in the number of rational points of a variety over a finite field. This project is also concerned with such a topic. The main result of this research project is as follows.
Let X be a hyper-surface of degree d in n-space over a finite field F of q-elements. We obtained a bound for the number of F-rational points of X. This bound is depending only on d, q and n, and also linear in d. We named this bound elementary bound. Moreover, for each (q, n), there are three hyper-surfaces with different degrees each other that attain the elementary bound. Additionally, we have determined the all surfaces in 3-space that attain the elementary bound.
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| Free Research Field |
代数幾何
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