Lê Dũng Tráng has a paper "The Geometry of the Monodromy Theorem" (MR0541020 https://mathscinet.ams.org/mathscinet/article?mr=541020 for reference). It's a very nice paper, and in it Lê gives a very geometric proof of quasi-unipotence of monodromy.

The ending of the paper contains some interesting suggestions for further work, and I was wondering if anyone has thought them through. I'm most interested in a suggested proof of the rationality of the roots of Bernstein's $b$-polynomial.

Question. Since 1978, has anyone found a way to use Lê's carrousels to prove rationality of roots of the $b$-polynomial without appealing to the resolution of singularities, as Lê suggests should be possible?

(I'm also broadly interested in hearing about any further developments on this work; I'm a student currently trying to learn some of Lê and others' work in this area, although most of my reading so far has been papers from the 70's....)

Lê Dũng Tráng has a paper "The Geometry of the Monodromy Theorem" (MR0541020 https://mathscinet.ams.org/mathscinet/article?mr=541020 for reference). It's a nice paper, and in it Lê gives a geometric proof of quasi-unipotence of monodromy.

The ending of the paper contains some interesting suggestions for further work, and I was wondering if anyone has thought them through. I'm most interested in a suggested proof of the rationality of the roots of Bernstein's $b$-polynomial.

Question. Since 1978, has anyone found a way to use Lê's carrousels to prove rationality of roots of the $b$-polynomial without appealing to the resolution of singularities, as Lê suggests should be possible?

(I'm also broadly interested in hearing about any further developments on this work; I'm a student currently trying to learn some of Lê and others' work in this area, although most of my reading so far has been papers from the 70's....)

Lê Dũng Tráng has a paper "The Geometry of the Monodromy Theorem" (MR0541020 https://mathscinet-ams-org.proxy.uchicago.edu/mathscinet/article?mr=541020 for reference). It's a very nice paper, and in it Lê gives a very geometric proof of quasi-unipotence of monodromy.

The ending of the paper contains some interesting suggestions for further work, and I was wondering if anyone has thought them through. I'm most interested in a suggested proof of the rationality of the roots of Bernstein's $b$-polynomial.

Question. Since 1978, has anyone found a way to use Lê's carrousels to prove rationality of roots of the $b$-polynomial without appealing to the resolution of singularities, as Lê suggests should be possible?

(I'm also broadly interested in hearing about any further developments on this work; I'm a student currently trying to learn some of Lê and others' work in this area, although most of my reading so far has been papers from the 70's....)

Lê Dũng Tráng has a paper "The Geometry of the Monodromy Theorem" (MR0541020 https://mathscinet.ams.org/mathscinet/article?mr=541020 for reference). It's a very nice paper, and in it Lê gives a very geometric proof of quasi-unipotence of monodromy.

The ending of the paper contains some interesting suggestions for further work, and I was wondering if anyone has thought them through. I'm most interested in a suggested proof of the rationality of the roots of Bernstein's $b$-polynomial.

Question. Since 1978, has anyone found a way to use Lê's carrousels to prove rationality of roots of the $b$-polynomial without appealing to the resolution of singularities, as Lê suggests should be possible?

(I'm also broadly interested in hearing about any further developments on this work; I'm a student currently trying to learn some of Lê and others' work in this area, although most of my reading so far has been papers from the 70's....)