发表机构
The George Washington University; physics.msu.ru（乔治华盛顿大学; 莫斯科国立大学物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文证明复杂带状Toeplitz矩阵的行列式递推可由简单因子递推组装，通过平移因子和线性不等式系统分解，并给出线性时间构造方法，以五对角例验证。
AI 中文摘要
Toeplitz矩阵是指每条对角线上的元素都保持常数的矩阵。当仅有有限多条对角线非零时，逐渐增大的矩阵的行列式遵循一个固定的线性递推关系：每个新行列式是有限个前驱行列式的固定线性组合。我们探讨一个复杂的带状矩阵的递推能否由更简单因子的递推构建，并证明这是可行的。两个有限带状Toeplitz矩阵相乘在内部区域重现预期的乘积，仅在两个相对角附近出现差异。将因子相对于主对角线进行平移会重新分布这些边界差异，而不同的平移恰好对应于构成完整行列式递推的各部分。对于多个因子，所有允许的平移由有限个线性不等式系统描述，从而给出递推的系统性分解。这一观点还引出一种递归构造方法，该方法直接处理多项式系数，无需求解其根。当提供了有界次数因子的分解时，可以仅用与需输出的系数数量成线性关系的算术运算次数构造出有效的递推。一个五对角例子展示了如何从两个三对角Toeplitz因子及两个边界贡献组装出一个六阶递推。
英文摘要
We study how determinant recurrences of banded Toeplitz matrices behave when their Laurent symbols are multiplied. Two classical structures underlie the problem. Clean banded Toeplitz determinants have a root-product description of their recurrences, while a product of two finite Toeplitz sections differs from the finite section of the product by finite-rank corner corrections. We connect these structures by identifying multiplication-induced boundary defects with exterior-degree sectors of the determinant recurrence. To each polynomial core we associate recurrence polynomials for all exterior degrees. Multiplication of cores becomes a graded convolution via composed products, while complementary degrees are related by a scaled reciprocal duality. On the matrix side, one-sided factors produce a defect filtration whose successive pieces are exactly these sectors. For general two-sided factors, independently weighting the two corner corrections gives an interval filtration: increasing either defect order adds one adjacent sector, and the cumulative recurrences are generically minimal. The multiplication-induced same-corner minors used in these filtrations lie in the standard row-column state module, whereas cross-corner imbalance corresponds instead to Laurent recentering. Opposite Laurent recenterings realize the individual sectors directly, and for several factors the admissible recenterings form an explicit lattice polytope. The pentadiagonal case gives the basic $1+4+1$ decomposition. The resulting framework gives a factor-level description of how finite-section boundary effects generate the recurrence spectrum of structured banded Toeplitz products.
