  We link the study of positive quantum maps, block positive operators, and
entanglement witnesses with problems related to multivariate polynomials. For
instance, we show how indecomposable block positive operators relate to
biquadratic forms that are not sums of squares. Although the general problem of
describing the set of positive maps remains open, in some particular cases we
solve the corresponding polynomial inequalities and obtain explicit conditions
for positivity.
