The modular counterparts of Cayley's hyperdeterminants
The modular counterparts of Cayley's hyperdeterminants
复制标题
凯莱超行列式的模对应项
DOI：
10.1017/s0004972700031890
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发表时间：
1998
影响因子：
0.7
通讯作者：
D. Glynn
中科院分区：
文献类型：
--
作者：
D. Glynn
设H是PG（n，q）中的m次超曲面，q = ph，p素. (1)如果m < n + 1，则H有1个（mod p）点。(2)如果m = n + 1，H有1个（mod p）点惠Hp−1没有项我们给出了一些应用，包括PG（n，F）中n + 1次超曲面的广义Hasse不变量，有限射影空间的各种性质，特别是域特征p上超立方体A上任意（n + 1）r+2 =（n + 1）×. ×（n + 1）阵列的p-模不变量detp。这个不变量是乘法的，因为detp（AB）= detp（B），只要定义了两个数组A和B的乘积（或卷积），并且两个数组都不是一维向量。(If A是（n + 1）r+2，B是（n + 1）s+2，则AB是（n + 1）r+s+2。）不变量的几何意义是，在特征为p的有限域上，A从阵列的任何给定r + 1个方向上的r + 1个点到最终方向上的非零点的投影数为0（mod p）。等价地，A从r个点在任意给定的r个方向上到非奇异（n + 1）2矩阵的投影数为0（mod p）。历史方面的不变量理论和连接凯莱的超行列式Det的特征0领域提到。
Let H be a hypersurface of degree m in PG(n, q), q = ph, p prime. (1) If m < n + 1, H has 1 (mod p) points. (2) If m = n + 1, H has 1 (mod p) points ⇔ Hp−1 has no term We show some applications, including the generalised Hasse invariant for hypersurfaces of degree n + 1 in PG(n, F), various porperties of finite projective spaces, and in particular a p-modular invariant detp of any (n + 1)r+2 = (n + 1)×…×(n + 1) array on hypercube A over a field characteristic p. This invariant is multiplicative in that detp(AB) = detp(B), whenever the product (or convolution of the two arrays A and B is defined, and both arrays are not 1-dimensional vectors. (If A is (n + 1)r+2 and B is (n + 1)s+2, then AB is (n + 1)r+s+2.) The geometrical meaning of the invariant is that over finite fields of characteristic p the number of projections of A from r + 1 points in any given r + 1 directions of the array to a non-zero point in the final direction is 0 (mod p). Equivalently, the number of projections of A from r points in any given r directions to a non-singular (n + 1)2 matrix is 0 (mod p). Historical aspects of invariant theory and connections with Cayley's hyperdeterminant Det for characteristic 0 fields are mentioned.