Jim Buddenhagen writes: << This polynomial: x^4 - 48*x^3 - 12*x^2 - 33*x + 1613 has a root very near Pi. How can I know if this is unusual for the size of the coefficients? -- I think that it is. I suppose this problem is in the category you describe -- but sorry I don't know how to do it.

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So this of course got me wondering just how close this root is to pi. Mathematica gives this root to 20 places as x = 3.1415926535884241315 . To 20 places, Pi = 3.1415926535897932385 . The difference seems astonishingly low, to 6 places: 1.369107 x 10^(-12). But if we consider all monic integer polynomials up to degree 4 using coefficients of absolute value up to 1613, there are say (3227^4 = 108,441,586,233,841 polynomials to consider, which makes this sound less astonishing. --------------------------------------------------- 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? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele