Positivity of Legendrian Thom polynomials

TL;DR

This paper proves that Legendrian Thom polynomials have nonnegative coefficients in a specific basis, extending previous nonnegativity results from Lagrangian Thom polynomials, and provides bounds for these coefficients.

Contribution

It introduces a new basis in the ring of Legendrian characteristic classes ensuring nonnegativity of Thom polynomial coefficients, extending prior work on Lagrangian cases.

Findings

Legendrian Thom polynomials have nonnegative coefficients in the new basis.

The method uses a Lagrange Grassmann bundle on product of projective spaces.

Provides upper bounds for coefficients based on classical Thom polynomial positivity.

Abstract

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.