In an interesting article (available here), Timothy Chow proposes that a closed-form number be defined as an element of the smallest subfield of $\mathbb{C}$ that is closed under $\exp$ and a chosen branch of $\log$. It is fun to check that pretty much any number that you might accept as closed-form answer to a calculus problem belongs to this field.

He writes, "My hope is that this definition of closed-form expression for a number will become standard, and that many readers will be lured into working on the many attarctive open problems in this area." The bulk of the article relates his notion of closed-form numbers to standard conjectures in transcendental number theory, most notably Schanuel's conjecture.

My questions are

To what extent has this notion become accepted as standard?

Are there new results since the time of his writing?

There was a rekindling of interest in Schanuel's conjecture after Boris Zil'ber's categoricity results on algebraically closed exponential fields in characteristic zero. In what way has this changed the status of problems mentioned in Chow's article (if it has)?

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    $\begingroup$ This regain of interest in Schanuel's conjecture is also due to the fact that Macintyre and Wilkie proved that SC implies the decidability of $\mathbb{R}_{\exp}$. $\endgroup$ – Thierry Zell Sep 29 '11 at 17:00
  • $\begingroup$ @Thierry: that's right. This was also mentioned in Timothy's article (page 447). For those interested: en.wikipedia.org/wiki/Tarski%27s_exponential_function_problem $\endgroup$ – Todd Trimble Sep 29 '11 at 17:48
  • $\begingroup$ There is a better notion of "nice numbers" namely periods (of forms on algebraic varieties defined over $\bar{\mathbb{Q}}$). There is a conjecture of Grothendieck that all algebraic relations among periods come from algebraic geometry and there is a paper of Kontshevich (that I don't have a reference to handy) which explores the relation of periods with more elementary notions. These topics have been studied a lot. $\endgroup$ – Felipe Voloch Sep 29 '11 at 17:56
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    $\begingroup$ @Felipe: could you explain why "better"? My understanding is that while periods might be "nice", the set of periods might not be particularly nice, algebraically (e.g., is conjectured not to form a field). It's not clear to me why these would have any claim to a status of a reasonable notion of closed-form numbers. $\endgroup$ – Todd Trimble Sep 29 '11 at 18:30
  • $\begingroup$ @Todd: "Better" might be too strong a word. It's just that there is more structure to work with. $\endgroup$ – Felipe Voloch Sep 29 '11 at 20:13

There is also the recent paper by Borwein and Crandall, Closed Forms: What they are and why we care", to appear in the Notices of the AMS. He gives 7 different methods via which one can approach closed forms. Chow's notion is #4. For some strange reason, I am rather a fan of the approach they baptized diffeoclosed...

  • $\begingroup$ I've accepted this, because it seems to answer my first question pretty well (Chow's proposal hasn't been accepted as "the standard" yet). Thanks again. $\endgroup$ – Todd Trimble Feb 27 '13 at 16:44

MathSciNet reports 3 papers that refer to Chow's paper...

MR2454730 Bronstein, Manuel; Corless, Robert M.; Davenport, James H.; Jeffrey, D. J. Algebraic properties of the Lambert W function from a result of Rosenlicht and of Liouville. Integral Transforms Spec. Funct. 19 (2008), no. 9-10, 709–712.

MR2180867 Richardson, Daniel; Elsonbaty, Ahmed Counterexamples to the uniformity conjecture. Comput. Geom. 33 (2006), no. 1-2, 58–64.

MR1854340 (2002j:11079) Richardson, Daniel Multiplicative independence of algebraic numbers and expressions. Effective methods in algebraic geometry (Bath, 2000). J. Pure Appl. Algebra 164 (2001), no. 1-2, 231–245.


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