Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier... Unlike Atiyah's paper, Hironaka's paper does not have a historical overview that provides a vague image for non-expert to the field. [Kollar] has such a comment:

...This duality also makes it difficult to write a reasonable historical presentation and to correctly appreciate the contributions of various researchers. Each step ahead can be viewed as small or large, depending on whether we focus on the change in the ideas or on their effect. In some sense, all the results of the past forty years have their seeds in [Hir64].

[Hauser] is a relatively readable review on this problem to me. My interest is basically on (possible) constructive side of [Hironaka] since singularity algorithms are known to connect with coding algorithms via Grobner bases. But given the status and attention given to this paper and I'm not an expert in AG, I want to ask

Can anyone comment on whether Hironaka's technique LLUED and its globalized version GLUED (like he said in the end of [Hironaka]) also improves design of resolution algorithms?

"Our methodology of using leverage-up exponent-down (LUED) seems to be more significant than just proving the resolution of singularities..."[Hironaka]

Will such a technique be extended beyond the scope of algebraic geometry (to, say, algebraic topology/coding theory/projective geometry)?

[Kollar] Kollár, János. "Resolution of Singularities--Seattle Lecture." arXiv preprint math/0508332 (2005).

[Hironaka] Resolution of Singularties in Positive Characteristics: A proof of resolution of singularities in characteristic $p>0$ and all dimensions.

[Hauser]Hauser, Herwig. "On the problem of resolution of singularities in positive characteristic (Or: A proof we are still waiting for)." Bulletin of the American Mathematical Society 47.1 (2010): 1-30. http://homepage.univie.ac.at/herwig.hauser/Publications/Problem_PosChar.pdf

[Hu] Hu, Yi. "Resolution of Singularities in Arbitrary Characteristic", https://arxiv.org/abs/2203.03842 (This is a main update on this topic as of 2023.)

please see recent updates in comments

  • 7
    $\begingroup$ Has anybody read/tried to read this? Is it serious? $\endgroup$
    – Igor Rivin
    May 13, 2017 at 3:02
  • 9
    $\begingroup$ Would be pretty cool if correct (55 years after the characteristic 0 result). $\endgroup$
    – Igor Rivin
    May 13, 2017 at 3:05
  • 12
    $\begingroup$ 92 pages of "hard" theory and not a single one example of a variety in char $p$ to which the theory is applied... $\endgroup$
    – Libli
    May 13, 2017 at 21:56
  • 10
    $\begingroup$ Unlike Atiyah's arXiv posting, this is rather detailed, and has had at least two people proof read it, according to the acknowledgements. $\endgroup$
    – David Roberts
    May 15, 2017 at 6:48
  • 6
    $\begingroup$ @David Roberts Yes, this paper has had at least two people proof read it, according to the acknowledgements, but only in a very weak sense. The acknowledgements refer only to "proof-reading and misprints-checking" by Woo Yang Lee of Seoul National University and Tadao Oda of Tohoku University. Whatever that means, I don't think Hironaka is acknowledging or suggesting something like a substantive check a peer-reviewed journal referee would give such a paper. $\endgroup$ Jul 6, 2017 at 4:22


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