The Conjectures of Alon--Tarsi and Rota in Dimension Prime Minus One
The Conjectures of Alon--Tarsi and Rota in Dimension Prime Minus One
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阿隆猜想--素数减一维度中的塔尔西和罗塔
DOI：
10.1137/090773751
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
D. Glynn
中科院分区：
文献类型：
--
作者：
D. Glynn
方阵的Glynn超行列式$\det_p$（$p$ prime）的一个公式表明，将任意行和和为p-1$的正整数双随机矩阵分解为p-1$偶积置换矩阵的方法数减去奇积置换矩阵的方法数为1（mod $p$）。因此，$p-1$阶的偶数拉丁方的个数不等于该阶的奇数拉丁方的个数。因此，罗塔的基础猜想是真实的一个向量空间的维数为$p-1$在任何领域的特征零或$p$，和所有其他特点，除了可能是有限的数字。它还表明，有一个错误，在出版的证明，声称乘以已知的维度的权力2，并声称，数量的偶数拉丁广场是大于数量的奇数拉丁广场。现在，26是最小的未知情况下，罗塔的基础猜想，甚至维向量空间的领域是未解决的。
A formula for Glynn's hyperdeterminant $\det_p$ ($p$ prime) of a square matrix shows that the number of ways to decompose any integral doubly stochastic matrix with row and column sums $p-1$ into $p-1$ permutation matrices with even product, minus the number of ways with odd product, is 1 (mod $p$). It follows that the number of even Latin squares of order $p-1$ is not equal to the number of odd Latin squares of that order. Thus Rota's basis conjecture is true for a vector space of dimension $p-1$ over any field of characteristic zero or $p$, and all other characteristics except possibly a finite number. It is also shown where there is a mistake in a published proof that claimed to multiply the known dimensions by powers of two, and that also claimed that the number of even Latin squares is greater than the number of odd Latin squares. Now, 26 is the smallest unknown case where Rota's basis conjecture for vector spaces of even dimension over a field is unsolved.