Distinctness and Algebraic Degrees of Inverted Kloosterman Sums

@misc{kyselov2026distinctness,
  title = {{Distinctness and Algebraic Degrees of Inverted Kloosterman Sums}},
  author = {Danyil Kyselov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25443},
  eprint = {2610.25443},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25443},
}

TY  - UNPB
TI  - Distinctness and Algebraic Degrees of Inverted Kloosterman Sums
AU  - Danyil Kyselov
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25443
UR  - https://arxiv.org/abs/2610.25443
AB  - We study the distinctness and algebraic degrees of inverted Kloosterman sums over finite extensions of 𝔽_p. For each d≥ 2, we prove that the (d−1)-dimensional sums over 𝔽_p^2 are equal precisely when their parameters are Frobenius conjugate, provided that p≡ 1pmod d, p>2d+1, and p does not divide the nonzero integer 2(d+1)^d−binom2dd^2. For two-dimensional sums over 𝔽_p^r, the same conclusion holds for 2≤ r≤ 5, p≡ 1pmod 3, and p>3r+1. In both cases we determine the algebraic degree of every sum in terms of the minimal polynomial of its parameter. The proof combines a cyclic-carry analysis of a Gauss-sum expansion with the recovery of power traces from a formal logarithm.
ER  -