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Vanishing Ideals of Projective Spaces over Finite Fields and a Projective Footprint Bound

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Abstract

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 Mercier and Rolland. 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.

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Acknowledgements

Peter Beelen would like to thank IIT Bombay where large parts of this work were carried out when he was there in January 2017 as a Visiting Professor. Sudhir Ghorpade would like to thank the Technical University of Denmark for a visit of 11 days in June 2017 when some of this work was done.

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Author notes

  1. Mrinmoy Datta

    Present address: Department of Mathematics and Statistics, UiT-The Arctic, University of Norway, N-9037, Tromsø, Norway

Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK, 2800, Kgs. Lyngby, Denmark

    Peter Beelen & Mrinmoy Datta

  2. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India

    Sudhir R. Ghorpade

Authors
  1. Peter Beelen
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  2. Mrinmoy Datta
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  3. Sudhir R. Ghorpade
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Corresponding author

Correspondence to Peter Beelen.

Additional information

Peter Beelen was supported by the Danish Council for Independent Research (Grant No. DFF–4002-00367). Mrinmoy Datta was supported by the Danish Council for Independent Research (Grant No. DFF–6108-00362), and he is currently supported by the Research Council of Norway (Project No. 280731). Sudhir Ghorpade is partially supported by IRCC Award grant 12IRAWD009 from IIT Bombay

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Beelen, P., Datta, M. & Ghorpade, S.R. Vanishing Ideals of Projective Spaces over Finite Fields and a Projective Footprint Bound. Acta. Math. Sin.-English Ser. 35, 47–63 (2019). https://doi.org/10.1007/s10114-018-8024-7

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