General computation of Thom polynomials

Develop a general computational method that, given an invariant, locally closed singularity subset η ⊂ E(m,n) (such as a contact singularity η_Q or a Thom–Boardman type) of complex codimension d, outputs the Thom polynomial Tp(η) ∈ Z[a_1,…,a_m,b_1,…,b_n] (or equivalently in the quotient variables c_i defined by (1 + c_1 t + c_2 t^2 + …) = (1 + b_1 t + b_2 t^2 + …)/(1 + a_1 t + a_2 t^2 + …)) satisfying the degeneracy loci interpretation for all maps f: M^m → N^n.

Background

The paper surveys several partial techniques—interpolation, resolutions, degenerations, quotient constructions, and Hilbert schemes—that successfully compute Thom polynomials in specific cases.

Despite these tools, the author emphasizes that there is no known universal procedure to compute Tp(η) for an arbitrary singularity, highlighting a central gap in the field.