AI 中文总结
该研究探讨有理Bishop算子的有限纤维行列式，通过奇偶性机制、显式循环性准则等分析，得到相关行列式的零点成因、连分数间隙函数及精确次数等结果。
AI 中文摘要
我们研究有理Bishop算子的有限纤维行列式及其在无理参数循环性中的作用。本文分为两个主要部分：第一部分，对于常向量$f=1$，结式恒等式与离散傅里叶分解揭示了一般模轨道阶的行列式奇偶性机制：奇数分母在正基本单元上给出非负归一化行列式，而偶数分母时，唯一的实交替傅里叶模式是唯一能产生变号零点的因子。我们给出$(r,q)=(9,16)$的显式例子和解析无穷族$(r,q)=(3,6n-2)$，Grivaux的无零点行列式被确定为连续阶子族$D_{1,q}$，因此这些零点具体由非连续模排序导致；第二部分，我们证明了一个显式循环性准则，该准则不要求纤维行列式的全局非退化性或单调性。定量Remez估计控制小行列式集，截断逆由端点校正的Fejér多项式近似，显式连续性模将得到的有理逼近传递到无理参数。这为$f=1$以及更一般的满足$f(0)\neq0$的每个多项式$f$生成了完全显式的连分数间隙函数，该论证还给出了相应多项式向量纤维行列式的精确次数和首项系数。
英文摘要
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 $f(0)\ne 0$. The argument also gives the exact degree and leading coefficient of the corresponding polynomial-vector fibre determinants.
