How "hard" is it to maximize a polynomial function subject to linear constraints?
Your problem is NP-hard, even for polynomials of degree 2. 
The crucial reference is  


  Theodore Motzkin and Ernst Strauss  (1965)
  "Maxima for graphs and a new proof of a theorem of Turan"
  Canadian Journal of Mathematics 17, pp 533-540  


Motzkin and Strauss consider an undirected graph $G=(V,E)$ 
with vertex set $V=\{1,2,\ldots,n\}$. 
They show that the optimal objective value of the following optimization problem equals $1-1/\omega$, where $\omega$ is the clique number of $G$:

\begin{eqnarray*}
\max &amp;&amp;\sum_{ij\in E} x_ix_j \\[2ex]
s.t. &amp;&amp;\sum_{i\in V} x_i=1 \\[1ex]
&amp;&amp; 0\le x_i\le 1~~~ \text{ for all $i\in V$}
\end{eqnarray*}

Since computing the clique number is NP-hard, this implies the NP-hardness of maximizing a multivariate polynomial function subject to linear constraints.