Rational Bishop determinants and explicit cyclicity criteria
Abstract
We study finite-fibre determinants for rational Bishop operators and their role in cyclicity for irrational parameters. The paper has two main parts. First, for the constant vector f=1, a resultant identity and a discrete Fourier factorization reveal a determinant parity mechanism for general modular orbit order: odd denominators give a nonnegative normalized determinant on the positive fundamental cell, while for even denominators the unique real alternating Fourier mode is the only factor capable of producing a sign-changing zero. We give an explicit example at (r,q)=(9,16) and an analytic infinite family (r,q)=(3,6n−2). Grivaux's zero-free determinant is identified as the consecutive-order subfamily D1,q, so these zeros are caused specifically by nonconsecutive modular ordering. Second, we prove an explicit cyclicity criterion that does not require global nondegeneracy or monotonicity of the fibre determinant. A quantitative Remez estimate controls the small-determinant set; cutoff inverses are approximated by endpoint-corrected Fejer polynomials; and an explicit continuity modulus transfers the resulting rational approximants to irrational parameters. This produces a fully explicit continued-fraction gap function for f=1 and, more generally, for every polynomial f with f(0)≠ 0. The argument also gives the exact degree and leading coefficient of the corresponding polynomial-vector fibre determinants.
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BibTeX
@misc{ma2026rational,
title = {{Rational Bishop determinants and explicit cyclicity criteria}},
author = {Yicen Ma},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27946},
eprint = {2608.27946},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27946},
}
RIS
TY - UNPB
TI - Rational Bishop determinants and explicit cyclicity criteria
AU - Yicen Ma
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.27946
UR - https://arxiv.org/abs/2608.27946
AB - We study finite-fibre determinants for rational Bishop operators and their role in cyclicity for irrational parameters. The paper has two main parts. First, for the constant vector f=1, a resultant identity and a discrete Fourier factorization reveal a determinant parity mechanism for general modular orbit order: odd denominators give a nonnegative normalized determinant on the positive fundamental cell, while for even denominators the unique real alternating Fourier mode is the only factor capable of producing a sign-changing zero. We give an explicit example at (r,q)=(9,16) and an analytic infinite family (r,q)=(3,6n−2). Grivaux's zero-free determinant is identified as the consecutive-order subfamily D_1,q, so these zeros are caused specifically by nonconsecutive modular ordering. Second, we prove an explicit cyclicity criterion that does not require global nondegeneracy or monotonicity of the fibre determinant. A quantitative Remez estimate controls the small-determinant set; cutoff inverses are approximated by endpoint-corrected Fejer polynomials; and an explicit continuity modulus transfers the resulting rational approximants to irrational parameters. This produces a fully explicit continued-fraction gap function for f=1 and, more generally, for every polynomial f with f(0)≠ 0. The argument also gives the exact degree and leading coefficient of the corresponding polynomial-vector fibre determinants.
ER -