On polynomials of small range sum
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917831914815488 |
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| author | Kiss, Gergely Markó, Ádám Nagy, Zoltán Lóránt Somlai, Gábor |
| author_facet | Kiss, Gergely Markó, Ádám Nagy, Zoltán Lóránt Somlai, Gábor |
| contents | In order to reprove an old result of Rédei's on the number of directions determined by a set of cardinality $p$ in $\mathbb{F}_p^2$, Somlai proved that the non-constant polynomials over the field $\mathbb{F}_p$ whose range sums are equal to $p$ are of degree at least $\frac{p-1}{2}$. Here the summand in the range sum are considered as integers from the interval $[0,p-1]$. In this paper we characterise all of these polynomials having degree exactly $\frac{p-1}{2}$, if $p$ is large enough. As a consequence, for the same set of primes we re-establish the characterisation of sets with few determined directions due to Lovász and Schrijver using discrete Fourier analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_06136 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On polynomials of small range sum Kiss, Gergely Markó, Ádám Nagy, Zoltán Lóránt Somlai, Gábor Number Theory Combinatorics 51E15, 11T06 In order to reprove an old result of Rédei's on the number of directions determined by a set of cardinality $p$ in $\mathbb{F}_p^2$, Somlai proved that the non-constant polynomials over the field $\mathbb{F}_p$ whose range sums are equal to $p$ are of degree at least $\frac{p-1}{2}$. Here the summand in the range sum are considered as integers from the interval $[0,p-1]$. In this paper we characterise all of these polynomials having degree exactly $\frac{p-1}{2}$, if $p$ is large enough. As a consequence, for the same set of primes we re-establish the characterisation of sets with few determined directions due to Lovász and Schrijver using discrete Fourier analysis. |
| title | On polynomials of small range sum |
| topic | Number Theory Combinatorics 51E15, 11T06 |
| url | https://arxiv.org/abs/2311.06136 |