# Length of surreal product

 Importance: Low ✭
 Author(s): Gonshor, Harry
 Subject: Combinatorics
 Keywords: surreal numbers
 Posted by: Lukáš Lánský on: April 7th, 2012
Conjecture   Every surreal number has a unique sign expansion, i.e. function , where is some ordinal. This is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of as .

It is easy to prove that

?

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

## Bibliography

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

* indicates original appearance(s) of problem.

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 , but I'll find time to go through it thoroughly as it is most relevant for the matter.