Name
Abstract | ||
|---|---|---|
About two decades ago, Tsfasman and Boguslavsky conjectured a formula for 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. 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 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. All these results are directly applicable to the determination of the maximum number of points on sections of Veronese varieties by linear subvarieties of a fixed dimension, and also the determination of generalized Hamming weights of projective Reed-Muller codes. |
Year | DOI | Venue |
|---|---|---|
2017 | 10.1090/proc/13863 | PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY |
Field | DocType | Volume |
Combinatorics,Classical orthogonal polynomials,Orthogonal polynomials,Mathematical analysis,Macdonald polynomials,Discrete orthogonal polynomials,Jacobi polynomials,Conjecture,Mathematics,Difference polynomials,Projective space | Journal | 146 |
Issue | ISSN | Citations |
4 | 0002-9939 | 0 |
PageRank | References | Authors |
0.34 | 0 | 3 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Peter Beelen | 1 | 16 | 2.11 |
| Mrinmoy Datta | 2 | 0 | 0.34 |
| Sudhir R. Ghorpade | 3 | 80 | 12.16 |