论文标题
伊茨 - 塞尔伯斯坦猜想的ROOK理论
Rook Theory of the Etzion-Silberstein Conjecture
论文作者
论文摘要
2009年，Etzion和Siberstein提出了对矩阵线性最大维度的猜想，在有限场上，在Ferrers图上支持所有非零矩阵，并以给定的整数为界面。尽管在过去十年中已经建立了一些猜想的案例，但证明或反驳这仍然是一个广泛的公开问题。在本文中，我们对Etzion-Siberstein的猜想进行了新的研究，并研究了其与Rook理论的联系。我们的结果表明，这个开放问题背后的组合学与Garsia和Remmel定义的$ Q $ -Rook多项式的理论紧密相关。从传来的角度来看，我们就其对角线的基础性的$ q $ -Rook多项式的尾随程度给出了一个封闭的公式。本文采用的组合方法使我们能够使用非构造论证来建立Etzion-Silberstein猜想的一些新实例。我们还解决了大型有限领域的猜想的渐近版本，回答了当前的开放问题。
In 2009, Etzion and Siberstein proposed a conjecture on the largest dimension of a linear space of matrices over a finite field in which all nonzero matrices are supported on a Ferrers diagram and have rank bounded below by a given integer. Although several cases of the conjecture have been established in the past decade, proving or disproving it remains to date a wide open problem. In this paper, we take a new look at the Etzion-Siberstein Conjecture, investigating its connection with rook theory. Our results show that the combinatorics behind this open problem is closely linked to the theory of $q$-rook polynomials associated with Ferrers diagrams, as defined by Garsia and Remmel. In passing, we give a closed formula for the trailing degree of the $q$-rook polynomial associated with a Ferrers diagram in terms of the cardinalities of its diagonals. The combinatorial approach taken in this paper allows us to establish some new instances of the Etzion-Silberstein Conjecture using a non-constructive argument. We also solve the asymptotic version of the conjecture over large finite fields, answering a current open question.
