I wrote: << 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? . . .
. . . The difference [between Pi and the root} 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 . . . 108,441,586,233,841 polynomials to consider, which makes this sound less astonishing.
Another valid viewpoint might be to consider only monic integer polynomials with sum-of-absolute- coefficients <= 48 + 12 + 33 + 1613 = 1706. For K > 0, the volume V(K) in 4-space of the region where |c_1| + . . . + |c_4| <= K is about the number of lattice points therein. I get V(K) = 2^4 * K^4 / 4! = 2 K^4 / 15, so V(1706) = 1,129,418,361,346 to the nearest integer, or about 1.1 x 10^12, about a trillion. Together with the answer to this question: [QUESTION: What's a good estimate for (maximum root of P) - (minimum root of Q) over all real roots of monic integer polynomials P, Q of degree d whose sum-of-absolute-values- of-coefficients is <= K ???] -- call this difference MM(d;K) -- then we can say the average spacing between such roots is about avspace = MM(4;1706) / 10^12 and we can ask what the expected distance from the nearest root to pi would be; this would be roughly avspace/4. So if MM(4; 1706) is significantly bigger than 4, we can suddenly gasp in surprise over the closeness to pi of the root Jim mentioned. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele