标题： Cappell-Shaneson多项式 显示英文标题

标题： Cappell-Shaneson polynomials

摘要： S. Cappell 和 J. Shaneson 为每一个阶数为$n$的特殊性质的方阵（Cappell-Shaneson 矩阵）构造了一对在同伦$(n+1)$-球面上不等价的$(n-1)$-球面嵌入。 Cappell-Shaneson 多项式是 Cappell-Shaneson 矩阵的特征多项式。 在本文中，我们将 Cappell-Shaneson 多项式的部分定义解释为关于带符号倒数多项式和模素数约简的多项式的代数条件，并给出了所有次数为$4$和$5$的 Cappell-Shaneson 多项式的完整列表。 我们构造了若干个次数为$6$的 Cappell-Shaneson 多项式的无限序列。 显示英文摘要

摘要： S. Cappell and J. Shaneson constructed a pair of inequivalent embeddings of $(n-1)$-spheres in homotopy $(n+1)$-spheres for every square matrix of order $n$ with special properties (a Cappell-Shaneson matrix). A Cappell-Shaneson polynomial is the characteristic polynomial of a Cappell-Shaneson matrix. In this paper, we interpret part of the definition of Cappell-Shaneson polynomial as algebraic conditions on polynomials in terms of signed reciprocal polynomial and reduction modulo primes, and give complete lists of all Cappell-Shaneson polynomials of degrees $4$ and $5$. We construct several infinite series of Cappell-Shaneson polynomials of degree $6$.