对称双线性型子空间的秩相关维数界
Rank-related dimension bounds for subspaces of symmetric bilinear forms
摘要：
设V是域K上的n维向量空间，Symm（V）表示定义在V × V上的对称双线性型空间，M是Symm（V）的子空间.我们调查了各种假设的秩的元素在M中，导致合理的界限，为暗M。例如，如果M的每个非零元素都有奇数秩，并且r是M的元素的最大秩，则dim M至多是r（r+1）/2（因此dim M是有界的，与n无关）。这应该与Symm（V）包含维度n-1的子空间的简单观察形成对比，其中每个非零元素具有秩2。界限r（r+1）/2几乎肯定太大了，而界限r似乎是合理的，当K是有限的时，这是真的。我们还表明，dimM是最r$时，K是任何领域的特征2。最后，假设n= 2 r，其中r是奇数，并且M的每个非零元素的秩是r或n。我们证明了如果K具有特征2，则dim M至多为3r。此外，如果dimM =3r，我们得到有趣的子空间分解的M和V有关的蔓延，伪弧和伪椭圆。如果K有r次的扩域，则存在这样的子空间M的例子。
英文摘要：
Let V be a vector space of dimension n over a field K and let Symm(V) denote the space of symmetric bilinear forms defined on V x V. Let M be a subspace of Symm(V). We investigate a variety of hypotheses concerning the rank of elements in M that lead to reasonable bounds for dim M. For example, if every non-zero element of M has odd rank, and r is the maximum rank of the elements of M, then dim M is at most r(r+1)/2 (thus dim M is bounded independently of n). This should be contrasted with the simple observation that Symm(V) contains a subspace of dimension n-1 in which each non-zero element has rank 2. The bound r(r+1)/2 is almost certainly too large, and a bound r seems plausible, this being true when K is finite. We also show that dim M is at most r$ when K is any field of characteristic 2. Finally, suppose that n=2r, where r is an odd integer, and the rank of each non-zero element of M is either r or n. We show that if K has characteristic 2, then dim M is at most 3r. Furthermore, if dim M=3r, we obtain interesting subspace decompositions of M and V related to spreads, pseudo-arcs and pseudo-ovals. Examples of such subspaces M exist if K has an extension field of degree r.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
