Proof of a Conjecture of Segre and Bartocci on Monomial Hyperovals in Projective Planes
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arXiv
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| Format: | Preprint |
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2010
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| _version_ | 1866909184230948864 |
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| author | Hernando, Fernando McGuire, Gary |
| author_facet | Hernando, Fernando McGuire, Gary |
| contents | The existence of certain monomial hyperovals $D(x^k)$ in the finite Desarguesian projective plane $PG(2,q)$, $q$ even, is related to the existence of points on certain projective plane curves $g_k(x,y,z)$. Segre showed that some values of $k$ ($k=6$ and $2^i$) give rise to hyperovals in $PG(2,q)$ for infinitely many $q$. Segre and Bartocci conjectured that these are the only values of $k$ with this property. We prove this conjecture through the absolute irreducibility of the curves $g_k$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1010_3965 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | Proof of a Conjecture of Segre and Bartocci on Monomial Hyperovals in Projective Planes Hernando, Fernando McGuire, Gary Combinatorics 11T06, 11T71 The existence of certain monomial hyperovals $D(x^k)$ in the finite Desarguesian projective plane $PG(2,q)$, $q$ even, is related to the existence of points on certain projective plane curves $g_k(x,y,z)$. Segre showed that some values of $k$ ($k=6$ and $2^i$) give rise to hyperovals in $PG(2,q)$ for infinitely many $q$. Segre and Bartocci conjectured that these are the only values of $k$ with this property. We prove this conjecture through the absolute irreducibility of the curves $g_k$. |
| title | Proof of a Conjecture of Segre and Bartocci on Monomial Hyperovals in Projective Planes |
| topic | Combinatorics 11T06, 11T71 |
| url | https://arxiv.org/abs/1010.3965 |