Contents - Student subdomain for University of Bath

A.2. USEFUL ESTIMATES 213
centring the polynomial first is almost always a good idea. Similarly re-balancing
changes the separation bound to 5.42 × 10 −112 , which means 6.52 × 10 −113 for
the original polynomial.
The monic polynomials of degree 7 with maximum root separation and all
roots in the unit circle are:
six complex x 7 − 1, with discriminant −823543, norm √ 2 and root bounds
[2.0, 1.383087554, 2.0]
and root separation bound (proposition 75) √ 3
56 ≈ 3.09 × 10−2 (true value
0.868);
four complex x 7 −x, with discriminant 6 6 = 46656, norm √ 2 and root bounds
[2.0, 1.383087554, 2.0]
and root separation bound (proposition 75) 27√ 21
18607
(true value 1);
≈ 7.361786385 × 10−3
two complex x 7 − 1 4 x5 − x 3 + 1 −50625
4
x, with discriminant
4096
≈ −12.36, norm
√
1
4 34 ≈ 1.457737974 and root bounds
[2, 1.626576562, 2.0]
and root separation bound (proposition 75) 1880√ 21
82572791 ≈ 9.99 × 10−5 (true
value 1/2);
all real x 7 − 14
9 x5 + 49
81 x3 − 4 81
x, with discriminant3
10485760000
1853020188851841 ≈ 5.66 × 10−6 , (A.1)
norm 17
81
√
86 ≈ 1.946314993 and root bounds
[ 23 , 3.299831645, 2.494438258]
9
and root separation bound (proposition 75)
83980800
32254409474403781
√
3
√
7 ≈ 1.19×
10 −8 (true value 1/3).
If there are n + 1 equally-spaced real roots (call the spacing 1 for the time
being), then the first root contributes n! to ∏ ∏
i j≠i (α i − α j ), the second root
1!(n − 1)! and so on, so we have a total product of
n∏
i!(n − i)! =
i=0
(
n∏
n 2
∏
i! 2 = i!)
. (A.2)
i=0
3 We note how the constraint that all the roots be real forces the discriminant to be small.
i=0