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Beelen, Peter; Datta, Mrinmoy; Ghorpade, Sudhir R.

Vanishing ideals of projective spaces over finite fields and a projective footprint bound. (English) Zbl 1411.14024

Acta Math. Sin., Engl. Ser. 35, No. 1, 47-63 (2019).
Summary: We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by D.-J. Mercier and R. Rolland [J. Pure Appl. Algebra 124, No. 1–3, 227–240 (1998; Zbl 0899.13028)]. We show that these generators form a universal Gröbner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre’s inequality for the number of points of projective hypersurfaces over finite fields is included.

Cited in 6 Documents

MSC:

14G15 Finite ground fields in algebraic geometry
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
11T06 Polynomials over finite fields
11G25 Varieties over finite and local fields
14G05 Rational points

Keywords:

finite field; projective space; algebraic variety; vanishing ideal; Gröbner basis; footprint bound; projective hypersurface

Citations:

Zbl 0899.13028
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Full Text: DOI arXiv

References:

[1] Becker, T., Weispfenning, V.: Gröbner Bases: A Computational Approach to Commutative Algebra, Springer-Verlag, Berlin, 1998 · Zbl 0772.13010
[2] Beelen, P., Ghorpade, S. R., Høholdt, T.: Duals of affine Grassmann codes and their relatives. IEEE Trans. Inform. Theory, 58, 3843-3855 (2012) · Zbl 1365.94579 · doi:10.1109/TIT.2012.2187171
[3] Beelen, P., Datta, M.: Generalized Hamming weights of affine cartesian codes. Finite Fields Appl., 51, 130-145 (2018) · Zbl 1416.94077 · doi:10.1016/j.ffa.2018.01.006
[4] Beelen, P., Datta, M., Ghorpade, S. R.: Maximum number of common zeros of homogeneous polynomial equations over finite fields. Proc. Amer. Math. Soc., 146, 1451-1468 (2018) · Zbl 1428.14036 · doi:10.1090/proc/13863
[5] Beelen, P., Datta, M., Ghorpade, S. R.: A combinatorial approach to the number of solutions of systems of homogeneous polynomial equations over finite fields, arXiv:1807.01683 [math.AG] · Zbl 1428.14036
[6] Carvalho, C., Neumann, V. G. L., Lopez, H. H.: Projective nested cartesian codes. Bull. Braz. Math. Soc., 48, 283-302 (2017) · Zbl 1386.14099 · doi:10.1007/s00574-016-0010-z
[7] Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 3rd Ed., Springer, New York, 1992 · Zbl 0756.13017 · doi:10.1007/978-1-4757-2181-2
[8] Couvreur, A.: An upper bound on the number of rational points of arbitrary projective varieties over finite fields. Proc. Amer. Math. Soc., 144, 3671-3685 (2016) · Zbl 1369.11044 · doi:10.1090/proc/13015
[9] Datta, M., Ghorpade, S. R.: On a conjecture of Tsfasman and an inequality of Serre for the number of points on hypersurfaces over finite fields. Mosc. Math. J., 15, 715-725 (2015) · Zbl 1348.14069 · doi:10.17323/1609-4514-2015-15-4-715-725
[10] Datta, M., Ghorpade, S. R.: Number of solutions of systems of homogeneous polynomial equations over finite fields. Proc. Amer. Math. Soc., 145, 525-541 (2017) · Zbl 1369.14033 · doi:10.1090/proc/13239
[11] Fitzgerald, J., Lax, R. F.: Decoding affine variety codes using Gröbner bases. Des. Codes Cryptogr., 13, 147-158 (1998) · Zbl 0905.94027 · doi:10.1023/A:1008274212057
[12] Geil, O., Høholdt, T.: Footprints or generalized Bezout’s theorem. IEEE Trans. Inform. Theory, 46, 635-641 (2000) · Zbl 1001.94040 · doi:10.1109/18.825832
[13] Geil, O., Martin, S.: Relative generalized Hamming weights of q – ary Reed-Muller codes. Adv. Math. Commun., 11, 503-531 (2017) · Zbl 1418.94073 · doi:10.3934/amc.2017041
[14] Ghorpade, S. R.: A note on Nullstellensatz over finite fields, arXiv:1806.09489 [math.AC] · Zbl 1430.14054
[15] Ghorpade, S. R., Lachaud, G.: Number of solutions of equations over finite fields and a conjecture of Lang and Weil, in Number Theory and Discrete Mathematics, pp. 269-291, Birkhäuser, Basel, 2002 · Zbl 1080.11049
[16] González-Sarabia, M., Martínez-Bernal, J., Villarreal, R. H., et al.: Generalized minimum distance functions, arXiv.math:1707.03285v3 [math.AC] · Zbl 1430.13049
[17] Hartshorne, R.: Algebraic Geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York, 1977 · Zbl 0367.14001 · doi:10.1007/978-1-4757-3849-0
[18] Heijnen, P., Pellikaan, R.: Generalized Hamming weights of q – ary Reed-Muller codes. IEEE Trans. Inform. Theory, 44, 181-196 (1998) · Zbl 1053.94581 · doi:10.1109/18.651015
[19] Høholdt, T.: On (or in) Dick Blahut’s footprint, in Codes, Curves and Signals, pp. 3-9, Kluwer, Norwell, MA, 1998
[20] Joly, J. R.: Equations et variétés algébraiques sur un corps fini. Enseign. Math., 19, 1-117 (1973) · Zbl 0282.14005
[21] Mercier, D. J., Rolland, R.: Polynomes homogenes qui s’annulent sur l’espace projectif Pm(Fq). J. Pure Appl. Algebra, 124, 227-240 (1998) · Zbl 0899.13028 · doi:10.1016/S0022-4049(96)00104-1
[22] Serre, J. P.: Lettreà M. Tsfasman, Journées Arithmétiques (Luminy, 1989). Astérisque No. 198-200, 351-353 (1991) · Zbl 0758.14008
[23] Sørensen, A. B.: Projective Reed-Muller codes. IEEE Trans. Inform. Theory, 37, 1567-1576 (1991) · Zbl 0741.94016 · doi:10.1109/18.104317
[24] Terjanian, G.: Sur les corps finis. C. R. Acad. Sci. Paris Sér. A-B, 262, A167-A169 (1966) · Zbl 0136.31901
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