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208 APPENDIX A. ALGEBRAIC BACKGROUND
Corollary 18 Disc(fg) = Disc(f) Disc(g) Res(f, g) 2 .
Whichever way they are calculated, the resultants are often quite large. For
example, if the a i and b i are integers, bounded by A and B respectively, the
resultant is less than (n + 1) m/2 (m + 1) n/2 A m B n , but it is very often of this
order of magnitude (see section A.2). Similarly, if the a i and b i are polynomials
of degree α and β respectively, the degree of the resultant is bounded by mα+nβ.
A case in which this swell often matters is the use of resultants to calculate
primitive elements, which uses the following result.
Proposition 64 If α is a root of p(x) = 0, and β is a root of q(x, α) = 0, then
β is a root of Res y (y − p(x), q(x, y)).
A.2 Useful Estimates
Estimates of the sizes of various things crop up throughout computer algebra.
They can be essentially of three kinds.
• Upper bounds: X ≤ B.
• Lower bounds: X ≥ B.
• Average sizes: X “is typically” B. This may be a definitive average
result (but even here we have to be careful what we are averaging over:
the average number of factors of a polynomial is one, for example, but this
does not mean we can ignore factorisation), or some heuristic “typically
we find”.
Estimates are used throughout complexity theory, of course. “Worst case” complexity
is driven by upper bound results, while “average case” complexity is
driven by average results. They are also used in algorithms: most algorithms of
the ‘Chinese Remainder’ variety (section 4) rely on upper bounds to tell when
they have used enough primes/evaluation points. In this case, a better upper
bound generally translates into a faster algorithm in practice.
A.2.1
Matrices
How big is the determinant |M| of an n × n matrix M?
Notation √∑
27 If v is a vector, then |v| denotes the Euclidean norm of v,
|v
2
i |. If f is a polynomial, |f| denotes the Euclidean norm of its vector
of ceofficients.
Proposition 65 If M is an n × n matrix with entries ≤ B, |M| ≤ n!B n .
This bound is frequently used in algorithm analysis, but we can do better.
Proposition 66 [Hadamard bound H r ] If M is an n × n matrix whose rows
are the vectors v i , then |M| ≤ H r = ∏ |v i |, which in turn is ≤ n n/2 B n .