Computational problems for vector-valued quadratic forms
<|reference_start|>Computational problems for vector-valued quadratic forms: 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.<|reference_end|>
@article{bullo2002computational,
    title={Computational problems for vector-valued quadratic forms},
    author={Francesco Bullo, Jorge Cortes, Andrew D. Lewis, Sonia Martinez},
    journal={arXiv preprint arXiv:math/0204068},
    year={2002},
    archivePrefix={arXiv},
    eprint={math/0204068},
    primaryClass={math.AG cs.CC math.OC}
}