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I wanted to know if anyone understood the details of the paper

"Multisingularities, cobordisms, and enumerative geometry" available at the site

In particular does any one follow how the author actually got all the expressions for $S_{A_k}$ on page 47. I think I follow the details of how he got $S_{A_1}$. But how does he get $S_{A_1^2}$ for instance?

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Presumably Kazarian does. This isn't a great MO question as written, but it could be fine. Although there's a reasonable chance that Kazarian will come along and answer your question, you're much more likely to get an answer by including just enough here to pose the question semi-self-containedly. There are probably experts who can answer your question, haven't read Kazarian's paper (or not recently), and don't feel like working through 47 pages, whereas since you have already worked through them, you could summarize the definitions and results needed. See – Theo Johnson-Freyd Aug 20 '10 at 5:22

It's just a task in school math.: The Legendrean residual class R for A_1^2 is given by -u-3a_1 on page 44, and the recipe for substitution is noted on page 45: the answer is the coefficient of t^2 in the expansion of R.[M]. If your question means how he gets the residual classes, you have to learn about the contents, that would be more fruitful.

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