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 to 0.6 eV, we are able to fit
the data with a precision of a few percents up to k=0.3 h / Mpc without any free parameter.