2 be viewed as polynomials in g[x~, • • • X~l], where 9 denotes the
polynomial domain
Zp[xv].
Although
F~'
and F( are primitive as elements of
Zp[xl,
• .
. , x~], they need not be primitive as elements of 9[x~, • • • , x~_~]. Let
c~, c2, c,
F1, F2 , G, H1, H2 , f~ , f~ , g, h~ , h~ , O, G, lP~ , F~ , H~ ,
and/t~ be defined as in Algorithm
M. Now, however, all of the lower ease symbols denote polynomials in
a = Z~[x~],
while the upper ease symbols denote polynomials in ~l[xl, • • • , X~_l].
For any fixed b ~ Zp, let g~ denote the field of polynomials in ~l modulo the ir
reducible polynomial (x~  b). Since for polynomials f ~ ~, the quantity
f(x~)
mod (xv  b) is equal tof(b) ~ Z~, we see that ~ is precisely Z~.
Algorithm P is essentially identical to Algorithm M, except that v is replaced by
v 
1, Z is replaced by 9, the prime p ~ Z is replaced by the irreducible polynomial
(x~  b) ~ g, and Z, is replaced by :¢~ = Zp.
In this situation, (31) is replaced by
F(~)
F1 mod (x~
b~),
1
=
(49)
while the polynomials 0 (i), H~), and
Zp[xl , .
.. , x~l].
Furthermore, (34
~(1)
and when e = d, we have
for i = 1,
/~2 mod(
x,

bi),
/~(i)
2 satisfy (32) and (33) in gb~[Xl, "" • , x~i] =
) becomes
0 mod (x~ 
b~),
(50)
~(i)
__
=
• " , n, in place of (35).
q = II
(xv  bl),
i=l
0 mod (xv 
/71 mod (x~  bd,
(51)
/I~ mod (x~ 
bl),
Also, (36) becomes
(52)
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while G*, Hi*, and H2* [see (37)] are the unique polynomials (in ~[Xa , ""
, xv,])
with eoeffieients (in 9) of degree (in
Xv)
less than n, such that
G* ~
G (i) rood (xv 
bi),
H,*  //~) rood (x, 
(53)
H2*
~
" (i)
1/12 mod (x, 
b~),
fori=
1,.
..,n.
Now as soon as e = d, weseefrom (51)and (53) that (38 holds.
When we also achieve
n > , = max (0v (0), o~ (/it,), 0~ (/72) ),
(54)
where 0v denotes the degree in x,, it follows that (40) holds. To obtain the final
results, we then use (41) and (42) as in Algorithm ~[.
Although the preceding discussion is sufficient in principle to define Algorithm P,
the interested reader may find it instructive to compare the following detailed
description with the earlier presentation (Section 4.3) of Algorithm M.
(1)
If v
1, then F, and F'
=
'
2 are elements of 9 invoke Algorithm U to comput
G' = gcd(F,', F(), and return. Otherwise use Algorithm U to compute 0 = eont(Fl' )
c2 = cont(F2'), c = gcd(o, c2).
(2)
Set F~ =
Fl'/C,, F~ = F2'/c2.
(3)
Set fl = lc(F,), f~ = lc(F2), 0 = gcd(f,, f2).
(4)
Set n = 0, e = min (~(F,), O(F~)).
(5)
Set ~, = 0~(0) +
O~(F,), ~2 = 0~(0) + O~(F2),
~ = max(p,, P2). It follows
that h = 0~ (f,) = 0~ (G) + 0v (/4,), ~2 = 0~ (f~) = 0~ (G) + 0~. (/t2), and v > ,.
(6)
Let b be a new element of Z, such that
(x~ 
b) ~ fir2.
If Zp is exhausted,
then p is too small and the algorithm fails.
(7)
Set 0 = 0 mod(x~ 
b), IP~ = OF,
mod(x, 
b), F2 = 0F2 mod(x~  b).
(8)
Invoke Algorithm P reeursively to compute G = g" ged
(F~, F2), lq, = IPl/G,
and tq2 = F2/G, all in
9b[xl, "" ,
X~_l] =
Zp[xi,
... , xv ,]. These relations imply
thatle(G)
= g, andl)(G) > d.
(9)
If it(G) = 0, set G =
1, Hi =
Fx, He = F2, and skip to Step (15). If
i} (G) > e, go back to Step (6). If i} (G) < e, set n = 0, e = ~ (G).
(10)
Setn=
n+
1.
(11)
If n = 1, set q = p, G* = G, H,* = /q, , H2* = ~q~ . Otherwise, update the
quadruple
(q, G*, H~*, H2*)
to include
(p,
G, /7,, H~) by using the Chinese re
mainder algorithm (Section 4.8) (which in this ease is a form of interpolation [1,
p. 430]) with moduli mi = q and m~ =
Xv  b to extend (53) (coefficient by co
efficient), and then replaeing q by
q(x~  b)
to extend (52).
(12)
Ifn < ~, go baek to Step (6). Otherwise, we now know thatn > ~ > ~,so
(40) holds unless e > d. To exclude this unlikely possibility, it suffiees to prove the
relations
G'Hi*
= ff~ and
*
*
G H~
= ~0~, which hold modulo q by (33), (49), (52),
and (53).
(13)
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 Spring '13
 MRR
 Math, Coefficient, F~

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