http://openproblemgarden.org/op/length_of_surreal_product Comments for "Length of surreal product" en http://openproblemgarden.org/op/length_of_surreal_product#comment-7176 <p>Thank you! I wasn't aware of this paper. At first sight I think that the part you refer to establish the required result just for surreals in the form <img class="teximage" src="/files/tex/63170bb5cb327ad8abc66f4623deb436aed9f2e0.png" alt="$ r\cdot\omega^x $" />, but I'll find time to go through it thoroughly as it is most relevant for the matter.</p> Tue, 05 Jun 2012 23:03:49 +0200 Lukáš Lánský comment 7176 at http://openproblemgarden.org http://openproblemgarden.org/op/length_of_surreal_product#comment-7172 <p>I believe the proof for the conjectured statement was proven in the affirmative in the paper "Fields of Surreal Numbers and Exponentiation" by Dries and Ehrlich. Specifically, Lemma 3.3 on page 6 : http://www.ohio.edu/people/ehrlich/EhrlichvandenDries.pdf</p> <p>If this satisfies the conjecture adequately great, if not, let me know if you would like to work toward a solution together on something similar or related. </p> <p>Thanks.</p> <p>-Vincent Russo</p> Wed, 30 May 2012 19:11:29 +0200 vprusso comment 7172 at http://openproblemgarden.org http://openproblemgarden.org/op/length_of_surreal_product <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/gonshor_harry">Gonshor</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/combinatorics">Combinatorics</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Every <a href="http://en.wikipedia.org/wiki/surreal number">surreal number</a> has a unique sign expansion, i.e. function <img class="teximage" src="/files/tex/70df8f7b1ba4ff48d933bb62e8ec5290ad07a83b.png" alt="$ f: o\rightarrow \{-, +\} $" />, where <img class="teximage" src="/files/tex/edb7612a8d6e29df825595761e763255474053d0.png" alt="$ o $" /> is some ordinal. This <img class="teximage" src="/files/tex/edb7612a8d6e29df825595761e763255474053d0.png" alt="$ o $" /> is the <em>length</em> of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of <img class="teximage" src="/files/tex/5161d8e30b9db389fca68be55f99b5f9e0f8ea7c.png" alt="$ s $" /> as <img class="teximage" src="/files/tex/156ccbe910e8f543b887e2294bcd5a450e454caf.png" alt="$ \ell(s) $" />.</p> <p>It is easy to prove that</p> <p><img class="teximage" src="/files/tex/e29d154689d0812c7decc7a0cbba897fa7cff1c6.png" alt="$$ \ell(s+t) \leq \ell(s)+\ell(t) $$" /></p> <p>What about</p> <p><img class="teximage" src="/files/tex/f7a5e9620fcb2831dff47c2a9273a45be4656b3b.png" alt="$$ \ell(s\times t) \leq \ell(s)\times\ell(t) $$" /></p> <p>? </div> </tr></td> </table> </td> </tr> </table> Gonshor, Harry surreal numbers Combinatorics http://openproblemgarden.org/op/length_of_surreal_product#comment Sat, 07 Apr 2012 22:58:50 +0200 Lukáš Lánský 37416 at http://openproblemgarden.org