  Understanding the biasing between the clustering properties of halos and the
underlying dark matter distribution is important for extracting cosmological
information from ongoing and upcoming galaxy surveys. While on sufficiently
larges scales the halo overdensity is a local function of the mass density
fluctuations, on smaller scales the gravitational evolution generates non-local
terms in the halo density field. We characterize the magnitude of these
contributions at third-order in perturbation theory by identifying the
coefficients of the non-local invariant operators, and extend our calculation
to include non-local (Lagrangian) terms induced by a peak constraint. We apply
our results to describe the scale-dependence of halo bias in cosmologies with
massive neutrinos. The inclusion of gravity-induced non-local terms and,
especially, a Lagrangian $k^2$-contribution is essential to reproduce the
numerical data accurately. We use the peak-background split to derive the
numerical values of the various bias coefficients from the excursion set peak
mass function. For neutrino masses in the range $0\leq \sum_i m_{\nu_i} \leq
0.6$ eV, we are able to fit the data with a precision of a few percents up to
$k=0.3\, h {\rm \,Mpc^{-1}}$ without any free parameter.
