but okay a lot of the time the actual *reason* i wanna work with formal power series is... i wanna *estimate* a_n. And it turns out that you can read off lots of stuff about the growth of a_n from the analytic properties of the *function represented by the power series*...
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Conversation
this is dope af of course, but...
what the heck does it *mean*???
does the species story have anything to say about the fact that the growth rate of a_n is controlled by the poles of the analytic function i get by *evaluating* its power series at *complex numbers* are?
4/shrug
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ive wondered about this a lot and afaict the answer is "not a chance"
and yet...
idk
there's GOTTA BE SOMETHING COME ON
5/5 im literally crying
this is your shit right save me
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this is a good question and i've sort of wondered fruitlessly about similar questions. a special case that i think is already pretty hopeless is: is there in any reasonable sense a "bijective" or "categorical" way to understand the asymptotics of the fibonacci numbers?
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hmm yeah
okay so there is an explicit formula for the n'th fibonacci number. usually u get it by mucking around with the generating function, do a partial fraction expansion blah blah
everything up to this point seems like it could plausibly categorify nicely...
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iirc there's nice combinatorially explicit way to see how when u expand out the binet formula w/ a binomial expansion on each of the (1+/-sqrt{5})^n terms it spits out the fibonacci numbers. something about tilings i forget how it works but i swear there's a nice way...
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but the issue is like...
only one of the terms contributes to the growth rate.
And while it's easy to categorify something like x^2+x-1, and maybe plausible to categorify "some expression involving the roots of that thing which magically has all the radicals cancel"...
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i don't see how to get past that point..
like, how can we tell the difference between the bigger root and the littler one? which is the only thing that actually matters???
IDK :shrug:
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currently i think at some point you really have to actually Do Some Analysis and you can't get away with not talking about, like, the order and metric structures on the reals
but it'd be very cool if i was wrong about that somehow although i can't imagine how
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one thing about the purely algebraic approach you're taking is that you haven't distinguished working over the reals from working over the p-adics and over the p-adics neither of the roots are negligible i think?
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hmm that was a good tweet. something something adelic combinatorics. at some point you have to make a choice that corresponds to working over R or C and not over the p-adics