  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.
