<< Dan later asks a similar question regarding the difference between the largest and smallest real roots. For that, consider: 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). This polynomial has roots near -2 and N+1 suggesting the largest spread of real roots is around n+3. I did not try to prove these are best possible, but they may be.

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I see my writing was not clear, since I wasn't looking for the maximum difference between roots of *any one polynomial*. Rather, among *all* monic integer polynomials of degree <= d, with a constraint on the size of the coefficients -- say the absolute values of the coefficients sum to <= N -- Then, among the set R(d: N) of *all* real roots of these polynomials, let the maximum be max(d; N) and the minimum be min(d; N). Then I'm interested here in an estimate (or asymptotics) for MM(d: N) := max(d; N) - min(d; N). --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele