Contents - Student subdomain for University of Bath

A.2. USEFUL ESTIMATES 209
Corollary 19 If f and g are polynomials of degrees n and m respectively, then
Res(f, g) ≤ (m + n) (m+n)/2 |f| m |g| n .
Corollary 20 If f is a polynomial of degree n, then Disc(f) ≤ n n−1 |f| 2n−1 .
In practice, especially if f is not monic, it is worth taking rather more care over
this estimation.
Proposition 67 (Hadamard bound (columns)) If M is an n × n matrix
whose columns are the vectors v i , then |M| ≤ H c = ∏ |v i | ≤ n n/2 B n .
In practice, for general matrices one computes min(H r , H c ). While there are
clearly bad cases (e.g. matrices of determinant 0), the Hadamard bounds are
“not too bad”. As pointed out in [AM01], log(min(H r , H c )/|M|) is a measure
of the “wasted effort” in a modular algorithm for computing the determinant,
and “on average” this is O(n), with a variance of O(log n). It is worth noting
that this is independent of the size of the entries.
Proposition 67 has a useful consequence.
Corollary 21 If x is the solution to M.x = a, where M and a have integer
entries bounded by B and A respectively, then the denominators of x are bounded
by min(H r , H c ) and the numerators of x are bounded by n n/2 AB n−1 .
Proof. This follows from the linear algebra result that x = 1
|M|
adj(M).a, where
adj(M), the adjoint of M, is the matrix whose (i, j)th entry is the determinant
of the matrix obtained by striking row i and column j from M. The ith entry
of adj(M).a is then the determinant of the matrix obtained by replacing the ith
column of M by a, and we can apply Proposition 67 to this matrix.
A.2.2
Coefficients of a polynomial
Here we are working implicitly with polynomials with complex coefficients,
though the bounds will be most useful in the case of integer coefficients.
Notation 28 Let
f(x) =
n∑
a i x i = a n
i=0
n ∏
i=1
(x − α i )
be a polynomial of degree n (i.e. a n ≠ 0). Define the following measures of the
size of the polynomial f:
H(f) (often written ||f|| ∞ ), the height or 1-norm, is max n i=0 |a i|;
||f|| (often written ||f|| 2 ), the 2-norm, is √ ∑ n
i=0 |a i| 2 ;
L(f) (often written ||f|| 1 ), the length or 1-norm, is ∑ n
i=0 |a i|;
∏
M(f) , the Mahler measure of f, is |a n | |α i |.
|α i|>1