New results on Chow's notion of closed-form numbers? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T19:00:11Z http://mathoverflow.net/feeds/question/76779 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76779/new-results-on-chows-notion-of-closed-form-numbers New results on Chow's notion of closed-form numbers? Todd Trimble 2011-09-29T16:42:51Z 2011-09-29T20:07:05Z <p>In an interesting article (available <a href="http://www-math.mit.edu/~tchow/closedform.pdf" rel="nofollow">here</a>), 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. </p> <p>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 <a href="http://en.wikipedia.org/wiki/Schanuel%27s_conjecture" rel="nofollow">Schanuel's conjecture</a>. </p> <p>My questions are </p> <blockquote> <p>To what extent has this notion become accepted as standard? </p> <p>Are there new results since the time of his writing? </p> </blockquote> <p>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)? </p> http://mathoverflow.net/questions/76779/new-results-on-chows-notion-of-closed-form-numbers/76783#76783 Answer by Gerald Edgar for New results on Chow's notion of closed-form numbers? Gerald Edgar 2011-09-29T17:48:14Z 2011-09-29T17:48:14Z <p>MathSciNet reports 3 papers that refer to Chow's paper...</p> <blockquote> <p>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.</p> <p>MR2180867 Richardson, Daniel; Elsonbaty, Ahmed Counterexamples to the uniformity conjecture. Comput. Geom. 33 (2006), no. 1-2, 58–64.</p> <p>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.</p> </blockquote> http://mathoverflow.net/questions/76779/new-results-on-chows-notion-of-closed-form-numbers/76796#76796 Answer by Jacques Carette for New results on Chow's notion of closed-form numbers? Jacques Carette 2011-09-29T20:07:05Z 2011-09-29T20:07:05Z <p>There is also the recent paper by Borwein and Crandall, <a href="http://carma.newcastle.edu.au/jon/closed-form.pdf" rel="nofollow">Closed Forms: What they are and why we care"</a>, to appear in the <em>Notices of the AMS</em>. 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 <em>diffeoclosed</em>...</p>