## Prove my lemma, get acknowledged in a paper!

This will be a little experiment, in which the collaborative mathematics advocated by Timothy Gowers and others combines with my own frustration and laziness. If it goes well, I might try it more in the future.

Let p be a complex polynomial of degree d. Suppose that |p(z)|≤1 for all z such that |z|=1 and |z-1|≥δ (for some small δ>0). Then what’s the best upper bound you can prove on |p(1)|?

**Note:** I can prove an upper bound of the form |p(1)|≤exp(δd)—indeed, that holds even if p can be a polynomial in both z and its complex conjugate (and is tight in that case). What *really* interests me is whether a bound of the form |p(1)|≤exp(δ^{2}d) is true.

**Update:** After I accepted Scott Morrison’s suggestion to post my problem at mathoverflow.net, the problem was solved 11 minutes later by David Speyer, using a very nice reduction to the case I’d already solved. Maybe I should feel sheepish, but I don’t—I feel grateful. I am now officially a fan of mathoverflow. Go there and participate!

Comment #1 December 14th, 2009 at 4:55 pm

Scott,

you should come and ask this over at http://mathoverflow.net/. Just make sure it doesn’t sound like a “homework problem”, or you might have your head bitten off.

Comment #2 December 14th, 2009 at 5:35 pm

Thanks, Scott! I just posted the question to mathoverflow; we’ll see what happens.

Comment #3 December 14th, 2009 at 6:19 pm

11 minutes later

Comment #4 December 15th, 2009 at 2:05 am

Wikipedia Math reference desk (http://en.wikipedia.org/wiki/WP:RDM) is another good place to ask this sort of question. I like WP better tham mathoverflow/stackoverflow because of its absence of “karma points” and other such dotcom devices.

Comment #5 December 15th, 2009 at 1:13 pm

Yeah, but he got a direct hit in 11 minutes.

Comment #6 December 16th, 2009 at 4:57 am

Anyway, “karma points” and whatnot is part of what makes MO so damn addictive (and hence successful.)

Comment #7 December 16th, 2009 at 6:11 pm

This is amazing. I saw this post shortly after it was posted and started thinking about the problem. I have no training beyond undergraduate-level math classes. I couldn’t even get to the exp bound myself. If I want to do this level of mathematics, where should I go for training? I’m going to read up on Chebyshev polynomials now.

Comment #8 December 19th, 2009 at 3:01 am

start reading math books and solving every problem you can find. there is no other way.