In another thread Dan Asimov asks: << QUESTION: By the way, is there some nice estimate for the maximum value among all real roots, of all monic integer polynomials of degree <= d, having coefficients of absolute value <= N ??? Call this M(d, N). Any good estimates, or asymptotic ones? >> I only looked a little at this, but the polynomial: x^d - N x^(d-1) - N x^(d-2) - ... - Nx - N (all terms negative but the first) has a real root very near N+1 and that may be as large as the real root can be. Note that subtracting 1 from this polynomial only changes the roots a little but the resulting polynomial has (x - (N+1)) as a factor. 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. Jim Buddenhagen