## Abstract

About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the

the maximum number of common zeros that r linearly independent homogeneous polynomials

of degree d in m + 1 variables with coefficients in a finite field with q elements can have in

the corresponding m-dimensional projective space over that finite field. Recently, it has been

shown by Datta and Ghorpade that this conjecture is valid if r is at most m + 1 and can

be invalid otherwise. Moreover a new conjecture was proposed for many values of r beyond

m + 1. In this paper, we prove that this new conjecture holds true for several values of r. In

particular, this settles the new conjecture completely when d = 3. Our result also includes

the positive result of Datta and Ghorpade as a special case. Further, we also determine the

maximum number of zeros in certain cases not covered by the earlier conjectures and results,

namely, the case of d = q − 1 and of d = q.

the maximum number of common zeros that r linearly independent homogeneous polynomials

of degree d in m + 1 variables with coefficients in a finite field with q elements can have in

the corresponding m-dimensional projective space over that finite field. Recently, it has been

shown by Datta and Ghorpade that this conjecture is valid if r is at most m + 1 and can

be invalid otherwise. Moreover a new conjecture was proposed for many values of r beyond

m + 1. In this paper, we prove that this new conjecture holds true for several values of r. In

particular, this settles the new conjecture completely when d = 3. Our result also includes

the positive result of Datta and Ghorpade as a special case. Further, we also determine the

maximum number of zeros in certain cases not covered by the earlier conjectures and results,

namely, the case of d = q − 1 and of d = q.

Original language | English |
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Journal | Proceedings of the American Mathematical Society |

Volume | 146 |

Issue number | 4 |

Pages (from-to) | 1451-1468 |

ISSN | 0002-9939 |

DOIs | |

Publication status | Published - 2017 |