 f_{2} =  f_{i}^{2}  (72) 
h  f  h_{2} B  (73) 
(2) > factor(x^105  1) (2) 2 4 3 2 (x  1)(x + x + 1)(x + x + x + x + 1) * 6 5 4 3 2 (x + x + x + x + x + x + 1) * 8 7 5 4 3 (x  x + x  x + x  x + 1) * 12 11 9 8 6 4 3 (x  x + x  x + x  x + x  x + 1) * 24 23 19 18 17 16 14 13 x  x + x  x + x  x + x  x 12 11 10 8 7 6 5 + x  x + x  x + x  x + x  x + 1 * 48 47 46 43 42 41 40 x + x + x  x  x  2x  x 39 36 35 34 33  x + x + x + x + x 32 31 28 26 24 22 20 + x + x  x  x  x  x  x 17 16 15 14 13 + x + x + x + x + x 12 9 8 7 6 5 2 + x  x  x  2x  x  x + x + x + 1 Type: Factored Polynomial IntegerWorse than that: for every B > 0 there exist infinitely many n such that there exists a polynomial h dividing x^{n}  1 and satisfying  h_{2} > B. See [vzGG99] for more details.

(75) 
f = f_{i}x^{i} = f_{n} (x  z_{i})  (76) 
We define the measure of the polynomial f by
M(f ) =  f_{n}  max(1,  z_{i}  )  (77) 
 z_{1}  ,...,  z_{k}  > 1 and  z_{k+1}  ,...,  z_{n}  1  (79) 
M(f ) =  f_{n}^{ . }z_{1}^{ ... }z_{k}   (80) 

(81) 

(82) 
 f_{1} =  f_{i}   (83) 
 f_{} = (  f_{i}  )  (84) 
 f_{}  f_{2}  f_{1} (n + 1)  f_{} and  f_{2} (n + 1)^{1/2}  f_{}  (85) 
 h_{2}  h_{1} 2^{m} M(h) 2^{m}  f_{2}  (86) 
h = h_{n} (x  u_{i})  (87) 
 h_{i}  M(h)  (88) 
 h_{2}  h_{1} M(h) 2^{m} M(f ) 2^{m}  f_{2} 2^{m}  (91) 
 g_{}  h_{}  g_{2}  h_{2}  g_{1}  h_{1} 2^{m+k}  f_{2} (n + 1)^{1/2} 2^{m+k}  f_{}  (92) 
 g_{1}  h_{1} 2^{m+k}  f_{2}.  (93) 
 g_{1} 2^{m} M(g) and  h_{1} 2^{k} M(h).  (94) 
M(g) M(h) M(f ).  (95) 
M(f )  f_{2}.  (96) 
 g_{1}  h_{1} 2^{m+k} M(g) M(h) 2^{m+k} M(f ) 2^{m+k}  f_{2}.  (97) 
 h_{} (n + 1)^{1/2} 2^{n}  f_{}  (98) 