Residue sums of Dickson polynomials over finite fields
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866910525250600960 |
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| author | Brazelton, Thomas Harrington, Joshua Litman, Matthew Wong, Tony W. H. |
| author_facet | Brazelton, Thomas Harrington, Joshua Litman, Matthew Wong, Tony W. H. |
| contents | Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over the distinct residues can sometimes be computed independent of the prime $p$; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the image set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_09119 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Residue sums of Dickson polynomials over finite fields Brazelton, Thomas Harrington, Joshua Litman, Matthew Wong, Tony W. H. Number Theory 11B39, 11C08 Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over the distinct residues can sometimes be computed independent of the prime $p$; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the image set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree. |
| title | Residue sums of Dickson polynomials over finite fields |
| topic | Number Theory 11B39, 11C08 |
| url | https://arxiv.org/abs/2103.09119 |