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(δ2d) is true.

Update: After I accepted Scott Morrison’s suggestion to post my problem at, 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!

8 Responses to “Prove my lemma, get acknowledged in a paper!”

  1. Scott Morrison Says:


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

  2. Scott Says:

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

  3. Scott Morrison Says:

    11 minutes later

  4. asdf Says:

    Wikipedia Math reference desk ( 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.

  5. Greg Kuperberg Says:

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

  6. harrison Says:

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

  7. bryce Says:

    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.

  8. dear bryce Says:

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