Expressing a fraction of two determinants as a determinant
Expressing a fraction of two determinants as a determinant
复制标题
将两个行列式的分数表示为行列式
DOI：
10.1145/1390768.1390790
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发表时间：
2008
影响因子：
1.4
通讯作者：
P. Koiran
中科院分区：
文献类型：
--
作者：
E. Kaltofen;P. Koiran
假设多项式f和g在K[x<sub>1</sub>，...，x<sub>r</sub>]分别是非奇异的m × m和n × n矩阵的行列式，其元素在K中为<$x<sub>1</sub>，...，<sub></sub>西河此外，假设h = f/g是K[x<sub>1</sub>，...，x<sub>r</sub>]。我们构造一个s x s矩阵C，其元素在K x<sub>1</sub>，...，<sub>xr</sub>，使得h = det（C）且s = γ（m+n）<sup>6</sup>，其中γ = O（1），如果K是无限域或对于有限域K = F{q}，有q个元素，则m = O（q），其中γ =（<sub>logqm</sub>）<sup>1+o（1）</sup>，如果q = o（m）.我们的构造利用了户田的偏斜电路和Malod和Portier的WSK电路的概念。我们的问题的动机是从Chow形式得到的结果公式。
此外，我们表明，部门可以从计算多项式的输入变量在一个足够大的领域内多项式公式大小增长的公式。
Suppose the polynomials f and g in K[x<sub>1</sub>,...,x<sub>r</sub>] over the field K are determinants of non-singular m x m and n x n matrices, respectively, whose entries are in K ∪ x<sub>1</sub>,...,x<sub>r</sub>. Furthermore, suppose h = f/g is a polynomial in K[x<sub>1</sub>,..., x<sub>r</sub>]. We construct an s x s matrix C whose entries are in K ∪ x<sub>1</sub>,...,x<sub>r</sub>, such that h = det(C) and s = γ (m+n)<sup>6</sup>, where γ = O(1) if K is an infinite field or if for the finite field K = F{q} with q elements we have m = O(q), and where γ = (log<sub>q</sub> m)<sup>1+o(1)</sup> if q = o(m). Our construction utilizes the notion of skew circuits by Toda and WSK circuits by Malod and Portier. Our problem was motivated by resultant formulas derived from Chow forms.
Additionally, we show that divisions can be removed from formulas that compute polynomials in the input variables over a sufficiently large field within polynomial formula size growth.
DOI：
10.1016/j.jco.2006.09.006
发表时间：
2006-08
期刊：
J. Complex.
影响因子：
--
作者：
Guillaume Malod;Natacha Portier
通讯作者：
Guillaume Malod;Natacha Portier