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
Conjecture   Every surreal number has a unique sign expansion, i.e. function $ f: o\rightarrow \{-, +\} $, where $ o $ is some ordinal. This $ o $ is the 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

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


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

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


*[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.


-Vincent Russo


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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