  We study Legendrian singularities arising in complex contact geometry. We
define a one-parameter family of bases in the ring of Legendrian characteristic
classes such that any Legendrian Thom polynomial has nonnegative coefficients
when expanded in these bases. The method uses a suitable Lagrange Grassmann
bundle on the product of projective spaces. This is an extension of a
nonnegativity result for Lagrangian Thom polynomials obtained by the authors
previously. For a fixed pecialization, other specializations of the parameter
lead to upper bounds for the coefficients of the given basis. One gets also
upper bounds of the coefficients from the positivity of classical Thom
polynomials (for mappings), obtained previously by the last two authors.
