Contents - Student subdomain for University of Bath

212 APPENDIX A. ALGEBRAIC BACKGROUND
Proposition 72 [Cau29, p. 123]
⎛ √
rb(f) ≤ max ⎝ n|a n−1|
,
a n
√
n|a n−2 |
, . . . , n−1
a n
√
n|a 1 |
, n
a n
⎞
n|a 0 |
⎠ .
a n
Proposition 73 [Knu98, 4.6.2 exercise 20]
⎛ √
rb(f) ≤ B = 2 max ⎝ |a n−1|
,
a n
√ √
|a n−2 | |a
, . . . , n−1 1 |
, n
a n
a n
Furthermore, there is at least one root of absolute value ≥ B/(2n).
⎞
|a 0 |
⎠ .
a n
Applied to W 20 , these propositions give respectively 1.38×10 19 , 4200 and 420. If
we centre the roots, to be −9.5, . . . , 9.5, the three propositions give respectively
2.17 × 10 12 , 400 and 40 (and hence the roots of W 20 are bounded by 2.17 × 10 12 ,
409.5 and 49.5). While the advantages of centrality are most prominent in
the case of proposition 71, they are present for all of them. There are in fact
improved bounds available in this (a n−1 = 0) case [Mig00].
If instead we re-balance W 20 so that the leading and trailing coefficients are
both 1, by replacing x by 20√ 20!x, then the bounds for this new polynomial are
6961419, 505.76 and 50.58 (and hence the roots of W 20 are bounded by 838284.7,
4200 and 420).
Proposition 74 ([Gra37]) If p(x) = p e (x 2 ) + xp o (x 2 ), i.e. p e (x 2 ) is the even
terms of p(x), then the roots of q(x) = p 2 e(x) − xp 2 o(x) are the squares of the
roots of p. 2
Applying this process to W 20 , then computing the three bounds, and squarerooting
the results, gives us bounds of 3.07 × 10 18 , 239.58 and 75.76. Repeating
the process gives 2.48 × 10 18 , 61.66 and 34.67. On the centred polynomial, we
get 2.73 × 10 12 , 117.05 and 37.01, and a second application gives 1.83 × 10 18 ,
30.57 and 17.19 (and hence root bounds for W 20 as 1.83×10 18 , 40.07 and 26.69).
This last figure is reasonably close to the true value of 20 [DM90].
A.2.4
Root separation
The key result is the following.
Proposition 75 ([Mah64]) sep(f) > √ 3| Disc(f)|n −(n+2)/2 |f| 1−n .
We note that the bound is zero if, and only if, the discriminant is zero, as it
should be, and this bound is unchanged if we multiply the polynomial by a
constant. The bound for W 20 is 7.27 × 10 −245 , but for the centred variant it
becomes 1.38×10 −113 . Should we ever need a root separation bound in practice,
2 For the history of the attribution to Graeffe, see [Hou59].