向量值二次型的计算问题
Computational problems for vector-valued quadratic forms
摘要：
给定两个实向量空间$U$和$V$，以及一个对称双线性映射$B: U\times U\to V$，设$Q_B$是它的二次映射$Q_B$。我们考虑的问题如下：（i）是否存在在多项式时间内可检验的判定$Q_B$满射的充要条件？(ii)如果$Q_B$是满射的，给定$v\in V$，是否有多项式时间算法来找到点$u\in Q_B^{-1}(v)$？(iii)是否有在多项式时间内可检验的充要条件来确定何时$B$是不定的？我们提出了另一种制定的问题，确定一个向量值二次型的图像的非投影Veronese曲面。这些问题的关系与控制理论中的几个有趣的问题进行了说明。
英文摘要：
Given two real vector spaces $U$ and $V$, and a symmetric bilinear map $B: U\times U\to V$, let $Q_B$ be its associated quadratic map $Q_B$. The problems we consider are as follows: (i) are there necessary and sufficient conditions, checkable in polynomial-time, for determining when $Q_B$ is surjective?; (ii) if $Q_B$ is surjective, given $v\in V$ is there a polynomial-time algorithm for finding a point $u\in Q_B^{-1}(v)$?; (iii) are there necessary and sufficient conditions, checkable in polynomial-time, for determining when $B$ is indefinite? We present an alternative formulation of the problem of determining the image of a vector-valued quadratic form in terms of the unprojectivised Veronese surface. The relation of these questions with several interesting problems in Control Theory is illustrated.
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。
