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214 APPENDIX A. ALGEBRAIC BACKGROUND
This is sequence A055209 [Slo07], which is the square of A000178.
If we now assume that the roots are equally-spaced in [−1, 1], then the
spacing is 2/n, we need to correct equation (A.2) by dividing by (n/2) n(n−1) :
call the result C n . C is initially greater than one, with C 3 = 65536
59049
≈ 1.11, but
C 4 = 81
1024 , C 5 = 51298814505517056
37252902984619140625 ≈ 0.001377042066, and C 6 as in (A.1).
While assuming equal spacing might seem natural, it does not, in fact, lead
to the largest values of the discriminant. Consider polynomials with all real
roots ∈ [−1, 1], so that we may assume the extreme roots are at ±1.
degree 4 Equally spaced roots, at ± 1 65536
3
, give a discriminant of
59049 ≈ 1.11,
whereas ± √ 1 5
gives 4096
3125
≈ 1.31, the optimum. The norms are respectively
√
182
9
≈ 1.4999 and √ 62
5
≈ 1.575.
degree 5 Equally spaced roots, at ± 1 2 and 0, give a discriminant of 81
√
1024 ≈
3
12288
0.079, whereas ±
7
and 0 gives
16807
≈ 0.73, the optimum. The norms
are respectively √ 42
4
≈ 1.62 and √ 158
7
≈ 1.796.
degree 6 Equally spaced roots, at ± 3 5 and ± 1 5 , give a discriminant of 51298814505517056
37252902984619140625 ≈
0.00138. Unconstrained solving for the maximum of the discriminant, using
Maple’s Groebner,Solve, starts becoming √ expensive, but if we assume
symmetry, we are led to choose roots at ± 147±42 √ 7
21
, with a discriminant
67108864
of
16209796869 ≈ 0.0041. The norms are respectively 2√ 305853
625
≈ 1.77 and
2 √ 473
21
≈ 2.07.
degree 7 Equally spaced roots, at ± 2 3 , ± 1 3 and 0, give a discriminant of 209715200000
5.66 √ · 10 −6 . Again assuming symmetry, we are led to choose roots at
± 495±66 √ 15
209715200000
33
and 0, which gives
5615789612636313 ≈ 3.73 · 10−5 . The norms
are respectively 17√ 86
81
≈ 1.95 and 2√ 1577
33
≈ 2.41,
degree 8 Equally spaced roots, at ± 5 7 , ± 3 7
and ± 1 7
, give a discriminant of
≈ 5.37 · 10 −9 . Assuming symmetry, we get roots at ± ≈ 0.87, ± ≈ 0.59
and ± ≈ 0.21, with a discriminant of ≈ 9.65 · 10 −8 . The norms are
respectively ≈ 2.15 and √ 2 √ 727171
429
≈ 2.81.
degree 9 Equally spaced roots, at ± 3 4 , ± 1 2 , ± 1 4
and 0, give a discriminant of
1.15·10 −12 . Assuming symmetry, we get roots at ±0.8998, ±0.677, ±0.363
and zero, with a discriminant of ≈ 7.03·10 −11 . The norms are respectively
√
5969546
1024
≈ 2.39 and ≈ 3.296.
If we now consider he case with two complex root, which may as well be at
x = ±i, we have the following behaviour.
degree 4 The maximal polynomial is x 4 −1, with discriminant −256 and norm
√
2. The bound is
√
6
216 ≈ 0.153.
degree 5 The maximal polynomial is x 5 −x, with discriminant −256 and norm
√
2. The bound is
4 √ 15
625 ≈ 0.0248.
5615789612636313 ≈