Localization formula conjecture for generating functions

Let QQ be a μ\mu-dimensional commutative nilpotent algebra, let kQk_Q be a generating function as in part (a) of the generating-function conjecture, and let e(Q,Mμ2)e(Q,\mathcal M_\mu^2) be the virtual Euler class defined for the ideal Mμ2\mathcal M_\mu^2. For a function in variables αi\alpha_i, write fαi:=zif|_{\alpha_i:=z_i} for the result of replacing αi\alpha_i by ziz_i, and define

Asymμ(f)=σSμε(σ)f(zσ(1),,zσ(μ)).\operatorname{Asym}_\mu(f)=\sum_{\sigma\in S_\mu}\varepsilon(\sigma)f(z_{\sigma(1)},\ldots,z_{\sigma(\mu)}).

Localization-form conjecture. If part (a) of the generating-function conjecture holds, then

Asymμ(kQ)=disμe(Q,Mμ2)αi:=zi.\operatorname{Asym}_\mu(k_Q)=\frac{\operatorname{dis}_\mu}{e(Q,\mathcal M_\mu^2)|_{\alpha_i:=z_i}}.

This conjecture proposes a connection between the residue and localization forms of Thom polynomials; it is conditional on the existence assertion in the preceding conjecture.

Sources & referencesView supporting material

Primary source

L. M. Fehér and R. Rimányi, “Thom series of contact singularities”, arXiv:0809.2925 (2010).

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