用消元法构造刚性矩阵
Using Elimination Theory to construct Rigid Matrices
摘要：
对于目标秩r，矩阵A的刚性是必须改变以确保改变的矩阵的秩至多为r的A的最小条目数。自从Valiant（1977）提出以来，矩阵的刚性和类似秩鲁棒性函数在电路复杂性、通信复杂性和学习复杂性方面有着广泛的应用。无限域上几乎所有的n × n矩阵的刚性都是（n-r）^2。当r = Omega（n）时，构造具有超线性刚性的显式矩阵无穷族是一个长期存在的问题。在本文中，我们构造了一个具有最大可能的复矩阵的无限族，即，(n-r)^2，刚性。在这个族中，一个n × n矩阵的元素是不同的单位原根，其阶数大约为exp（n^2 log n）。据我们所知，这是第一个家庭的具体（但不完全明确）矩阵具有最大的刚度和简洁的代数描述。我们的构造是基于多项式理想的消去理论。特别是，我们使用的结果消除理想的多项式的存在性与有效的程度上界（有效的Nullstellenitus）。利用初等代数几何，我们证明了至多k个刚性矩阵的仿射簇的维数恰好是n^2-（n-r）^2+k。最后，利用消元法检验刚性函数是否半连续。
英文摘要：
The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be changed to ensure that the rank of the altered matrix is at most r. Since its introduction by Valiant (1977), rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a long-standing open question to construct infinite families of explicit matrices even with superlinear rigidity when r = Omega(n). In this paper, we construct an infinite family of complex matrices with the largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in this family are distinct primitive roots of unity of orders roughly exp(n^2 log n). To the best of our knowledge, this is the first family of concrete (but not entirely explicit) matrices having maximal rigidity and a succinct algebraic description. Our construction is based on elimination theory of polynomial ideals. In particular, we use results on the existence of polynomials in elimination ideals with effective degree upper bounds (effective Nullstellensatz). Using elementary algebraic geometry, we prove that the dimension of the affine variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we use elimination theory to examine whether the rigidity function is semi-continuous.
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