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130 CHAPTER 4. MODULAR METHODS
Figure 4.9: Diagrammatic illustration of g.c.d.s in Z[x, y] (1)
Z[y][x] - - - - - - - - - - - - - - - - - - - - - - - - -> - Z[y][x]
k×reduce ↓
↑ interpret
& check
Z[y] y=v1 [x]
.
gcd
−→
.
gcd
Z[y] y=v1 [x]
Z[y] y=vk [x] −→ Z[y] y=vk [x] ⎪⎭
↑
using Algorithm 14/15
.
⎫
⎪⎬
C.R.T.
−→
Z[y][x]
∏ ′
(y − v 1 ) · · · (y − v k )
Figure 4.10: Diagrammatic illustration of g.c.d.s in Z[x, y] (2)
Z[y][x] - - - - - - - - - - - - - - - - - - - - - - - - -> - Z[y][x]
k×reduce ↓
↑ interpret
& check
Z p1 [y][x]
.
gcd
−→
.
gcd
Z p1 [y][x]
Z pk [y][x] −→ Z pk [y][x] ⎪⎭
↑
using Algorithm 17
.
⎫
⎪⎬
C.R.T.
−→
Z[y][x]
∏ ′
p 1 · · · p k
Should this happen, we actually know that all reductions so far are wrong,
because they are compatible in degree, but cannot be right. Hence we start
again.
Z p [y][x] Here we use an analogue of Figure 4.2 to reduce us to the case of Figure
4.7. The overall structure is shown in Figure 4.10.
Open Problem 6 (Which is the Better Route for Bivariate g.c.d.?) As
far as the author can tell, the question of which route to follow has not been systematically
explored. The initial literature [Bro71b, Algorithm M] and the more
recent survey [Lau82] assume the route in Figure 4.10. [Bro71b] explicitly assumes
that p is large enough that we never run out of values, i.e. that the
algebraic extension at the end of Figure 4.8 is never needed.
Implementations the author has seen tend to follow the route in Figure 4.9.
This is probably for pragmatic reasons: as one is writing a system one first
wishes for univariate g.c.d.s, so implements Algorithm 14 (or 15). Once one
has this, Figure 4.9 is less work.
There is a natural tendency to believe that Figure 4.10 is better, as all numbers
involved are bounded by p untl the very last Chinese Remainder calculations.