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Determining steady state behaviour of discrete monomial dynamical systems
Certain sextics with many rational points
Department of Mathematics, Shiga University of Medical Science, Seta Tsukinowa-cho, Otsu, Shiga, 520-2192 Japan |
We construct a family of sextics from the Wiman and Edge sextics. We find a curve over $\mathbb{F}_{5^7}$ attaining the Serre bound, and update $9$ entries of genus $6$ in manYPoints.org by computer search on these sextics.
References:
[1] |
W. Bosma and J. Cannon andC. Playoust,
The Magma algebra system. I. The user language, J. Symb. Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[2] |
W. L. Edge,
A pencil of four-nodal plane sextics, Math. Proc. Cambridge Philos. Soc., 89 (1981), 413-421.
doi: 10.1017/S0305004100058321. |
[3] |
A. Garcia, G. Güneri and H. Stichtenoth,
A generalization of the Giulietti--Korchmáros maximal curve, Adv. Geom., 10 (2010), 427-434.
doi: 10.1515/ADVGEOM.2010.020. |
[4] |
G. van der Geer, E. Howe, K. Lauter and C. Ritzenthaler,
Table of curves with many points available at http://www.manypoints.org |
[5] |
M. Giulietti, M. Montanucci and G. Zini,
On maximal curves that are not quotients of the Hermitian curve, Finite Fields Appl, 41 (2016), 72-88.
doi: 10.1016/j.ffa.2016.05.005. |
[6] |
E. Kani and M. Rosen,
Idempotent relations and factors of Jacobians, Math. Ann., 284 (1989), 307-327.
doi: 10.1007/BF01442878. |
[7] |
M. Q. Kawakita,
Wiman's and Edge's sextic attaining Serre's bound preprint, available at http://www.manypoints.org/upload/kawakita.pdf |
[8] |
M. Q. Kawakita, Wiman's and Edge's sextic attaining Serre's bound Ⅱ, in Algorithmic Arithmetic, Geometry, and Coding Theory, Amer. Math. Soc., 2015, 191–203.
doi: 10.1090/conm/637/12758. |
[9] |
K. Lauter,
Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields, J. Algebraic Geom., 10 (2001), 19-36.
|
[10] |
C. Moreno,
Algebraic Curves over Finite Fields Cambridge Univ. Press, 1991.
doi: 10.1017/CBO9780511608766. |
[11] |
A. Wiman,
Ueber eine einfache Gruppe von 360 ebenen Collineationen, Math. Ann., 47 (1896), 531-556.
doi: 10.1007/BF01445800. |
show all references
References:
[1] |
W. Bosma and J. Cannon andC. Playoust,
The Magma algebra system. I. The user language, J. Symb. Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[2] |
W. L. Edge,
A pencil of four-nodal plane sextics, Math. Proc. Cambridge Philos. Soc., 89 (1981), 413-421.
doi: 10.1017/S0305004100058321. |
[3] |
A. Garcia, G. Güneri and H. Stichtenoth,
A generalization of the Giulietti--Korchmáros maximal curve, Adv. Geom., 10 (2010), 427-434.
doi: 10.1515/ADVGEOM.2010.020. |
[4] |
G. van der Geer, E. Howe, K. Lauter and C. Ritzenthaler,
Table of curves with many points available at http://www.manypoints.org |
[5] |
M. Giulietti, M. Montanucci and G. Zini,
On maximal curves that are not quotients of the Hermitian curve, Finite Fields Appl, 41 (2016), 72-88.
doi: 10.1016/j.ffa.2016.05.005. |
[6] |
E. Kani and M. Rosen,
Idempotent relations and factors of Jacobians, Math. Ann., 284 (1989), 307-327.
doi: 10.1007/BF01442878. |
[7] |
M. Q. Kawakita,
Wiman's and Edge's sextic attaining Serre's bound preprint, available at http://www.manypoints.org/upload/kawakita.pdf |
[8] |
M. Q. Kawakita, Wiman's and Edge's sextic attaining Serre's bound Ⅱ, in Algorithmic Arithmetic, Geometry, and Coding Theory, Amer. Math. Soc., 2015, 191–203.
doi: 10.1090/conm/637/12758. |
[9] |
K. Lauter,
Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields, J. Algebraic Geom., 10 (2001), 19-36.
|
[10] |
C. Moreno,
Algebraic Curves over Finite Fields Cambridge Univ. Press, 1991.
doi: 10.1017/CBO9780511608766. |
[11] |
A. Wiman,
Ueber eine einfache Gruppe von 360 ebenen Collineationen, Math. Ann., 47 (1896), 531-556.
doi: 10.1007/BF01445800. |
5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 |
W | W | W | W | W | |||
31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 |
W | S | W | S | W | S | W | |
67 | 71 | 73 | 79 | 83 | 89 | 97 | 101 |
S | W | W | W | W | S | W | |
103 | 107 | 109 | 113 | 127 | 131 | 137 | 139 |
W | W | E | S | W | W | S | W |
149 | 151 | 157 | 163 | 167 | 173 | 179 | 181 |
E | S | S | W | S | W | S | |
191 | 193 | 197 | 199 | ||||
W | W | S | W |
5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 |
W | W | W | W | W | |||
31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 |
W | S | W | S | W | S | W | |
67 | 71 | 73 | 79 | 83 | 89 | 97 | 101 |
S | W | W | W | W | S | W | |
103 | 107 | 109 | 113 | 127 | 131 | 137 | 139 |
W | W | E | S | W | W | S | W |
149 | 151 | 157 | 163 | 167 | 173 | 179 | 181 |
E | S | S | W | S | W | S | |
191 | 193 | 197 | 199 | ||||
W | W | S | W |
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