Contents - Student subdomain for University of Bath

A.2. USEFUL ESTIMATES 211
Proposition 69 is still valid.
Corollary 23 Hence
A.2.3
|a i1,...,i n
| ≤
Roots of a polynomial
( )( ) ( )
d1 d2 dn
· · · ||f||.
i 1 i 2 i n
Several questions can be asked about the roots of a univariate polynomial. The
most obvious ones are how large/small can they be, but one is often interested in
how close they can be. These questions are often asked of the real roots (section
3.5.4), but we actually need to study all roots, real and complex. The distinction
between real and complex roots is pretty fine, as shown in the following example.
Example 15 (Wilkinson Polynomial [Wil59]) Let W 20 have roots at −1 ,
−2 , . . . , −20, so that W 20 = (x+1)(x+2) . . . (x+20) = x 20 +210x 19 +· · ·+20!.
Consider now the polynomial W 20 (x) + 2 −23 x 19 . One might expect this to have
twenty real roots close to the original ones, but in fact it has ten real roots, at
approximately −1, −2, . . . −7, −8.007, −8.917 and −20.847, and five pairs of
complex conjugate roots, −10.095±0.6435i, −11.794±1.652i, −13.992±2.519i,
−16.731 ± 2.813i and −19.502 ± 1.940i.
The discriminant of W 20 is 2.74 × 10 275 , which would seem well away from zero.
However, the largest coefficient of W 20 is 1.38×10 19 , and of W 20 ′ is −3.86×10 19 .
The norms of W 20 and W 20 ′ are 2.27 × 10 19 and 6.11 × 10 19 , so corollary 20 gives
a bound of 4.43 × 10 779 for Disc(W 20 ), and a direct application of corollary
19 gives 3.31 × 10 763 . Hence the discriminant of W 20 is “much smaller than
it ought to be”, and W 20 is “nearly not square-free”. Put another way, the
Sylvester matrix for the discriminant is very illconditioned (this was in fact
Wilkinson’s original motivation for constructing W 20 ): the discrepancy between
the actual determinant and corollary 19 is 489 decimal digits, whereas [AM01]
would lead us to expect about 17.
Notation 29 Let f ∈ C[x] = ∑ n
i=0 a ix i , and let the roots of f be α 1 , . . . , α n ,
and define
rb(f) = max
1≤i≤n |α i|,
sep(f) =
min |α i − α j |.
1≤i