Length of surreal product

Importance: Low ✭
Author(s): Gonshor, Harry
Subject: Combinatorics
Keywords: surreal numbers
Recomm. for undergrads: yes
Posted by: Lukáš Lánský
on: April 7th, 2012

\begin{conjecture} Every \Def{surreal number} has a unique sign expansion, i.e. function $f: o\rightarrow \{-, +\}$, where $o$ is some ordinal. This $o$ is the \emph{length} of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of $s$ as $\ell(s)$.

It is easy to prove that

$$ \ell(s+t) \leq \ell(s)+\ell(t) $$

What about

$$ \ell(s\times t) \leq \ell(s)\times\ell(t) $$

? \end{conjecture}

This is strongly conjectured to be true by Gonshor in [Gon86]. There is an easy way to prove that

$$ \ell(s\times t) \leq 3^{\ell(s)+\ell(t)} $$

Bibliography

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

*[Gon86] Harry Gonshor, An Introduction to the Theory of Surreal Numbers, Cambridge University Press, Cambridge, 1986.


* indicates original appearance(s) of problem.

Proof Already Exists?

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

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.

Thanks.

-Vincent Russo

Maybe!

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 $r\cdot\omega^x$, but I'll find time to go through it thoroughly as it is most relevant for the matter.

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