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Topic: This Week's Finds in Mathematical Physics (Week 236)
Replies: 29   Last Post: Aug 24, 2006 9:00 AM

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 john baez Posts: 460 Registered: 12/6/04
Re: This Week's Finds in Mathematical Physics (Week 236)
Posted: Aug 1, 2006 1:11 AM

<david.corfield@tuebingen.mpg.de> wrote:

>> Maybe you meant Motzkin *trees*.

>No, I did mean paths...

Okay, whoops - I'd forgotten all about that!

>But of course there are loads of combinatorial interpretations,
>including the trees you mention (which was actually the way I found the
>link between Leinster and Fiore's construction and the Motzkin
>numbers.)

Sorry, I even forgot your role in this. It winds up that trees
are more useful than paths, because if we have any polynomial
fixed-point equation with natural number coefficients, like

X = 349X^7 + 3X^4 + 2

we can interpret it as defining a set of "colored trees".
To do this, we interpret the equal sign as an isomorphism.

For example, the above equation defines the set X of planar,
rooted, finite trees where the root is colored in one of 2
ways, and each node either has 4 branches coming out of it
(in which case this node is colored in one of 3 ways) or
7 branches coming out of it (in which case it's colored in
one of 349 ways).

By the work of Schanuel, Gates, Fiore and Leinster, it makes
sense to assign a cardinality to this set which is a root of
the above polynomial.

A good example is the "Golden Object" discovered by Robin Houston:

http://math.ucr.edu/home/baez/week203.html

The golden number is usually defined by the equation

G^2 = G + 1 (1)

which is not of the fixed-point form described above.
However, Houston made the substitution

G = H + 2 (2)

and notes that

H = H^2 + 4H + 1 (3)

is of the desired fixed-point form. So, H is a set of colored
trees and G is the disjoint union of H and the 2-element set.

So, we can think of G as a set of "colored trees" of a slightly
more general type. (Houston gives another interpretation.)

Anyway, using this sort of trick we might able to cook up a
category of colored trees containing the category of finite
sets, but also objects having cardinality equal to any algebraic
number! This would be a candidate for a categorified version
of the algebraic closure of Q.