On Thu, Sep 18, 2008 at 10:55 PM, Dan Asimov <dasimov@earthlink.net> wrote: [...snip...]
I see my writing was not clear, since I wasn't looking for the maximum difference between roots of *any one polynomial*. [...]
Sorry for misunderstanding the question. I'm afraid the I don't know the answer to your question, but, irrespective of that, let me still ask two (related) questions: (1) Among *all* monic integer polynomials of degree = d, with a constraint on the size of the coefficients -- namely, the absolute value of each coefficient is <= N -- is it the case that this polynomial: x^d - N x^(d-1) - N x^(d-2) - ... - Nx - N (all terms negative but the first) has the largest real root? (2) Among *all* monic integer polynomials of degree = d, with a constraint on the size of the coefficients -- namely, the absolute value of each coefficient is <= N -- is it the case that this polynomial: x^d - N x^(d-1) - N x^(d-2) + N x^(d-3) - N x^(d-4) + ... (first 2 terms after leading term are negative after which terms alternate in sign; last term is + or - N depending on parity of d). has the largest difference between its biggest and smallest real roots? Jim