# HG changeset patch
# User David Loeffler <d.loeffler.01@cantab.net>
# Date 1285673392 3600
# Node ID 62945bccdb3bbeb4531b7d7ac63668491a4e2e76
# Parent ae11a4bd7bec1c4f77bd1e7cf64fde8745d9ba36
#9359: fix duplicate citations in reference manual
diff r ae11a4bd7bec r 62945bccdb3b sage/rings/number_field/totallyreal.pyx
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12  12  Algorithm 
13  13   
14  14  
15   We use Hunter's algorithm ([C]_, Section 9.3) with modifications 
16   due to Takeuchi [T]_ and the author [V]_. 
 15  We use Hunter's algorithm ([Cohen2000]_, Section 9.3) with modifications 
 16  due to Takeuchi [Takeuchi1999]_ and the author [Voight2008]_. 
17  17  
18  18  We enumerate polynomials `f(x) = x^n + a_{n1} x^{n1} + \dots + a_0`. 
19  19  Hunter's theorem gives bounds on `a_{n1}` and `a_{n2}`; then given 
20  20  `a_{n1}` and `a_{n2}`, one can recursively compute bounds on `a_{n3}, 
21  21  \dots, a_0`, using the fact that the polynomial is totally real by 
22  22  looking at the zeros of successive derivatives and applying 
23   Rolle's theorem. See [T]_ for more details. 
 23  Rolle's theorem. See [Takeuchi1999]_ for more details. 
24  24  
25  25  Examples 
26  26   
… 
… 

68  68  References 
69  69   
70  70  
71   .. [C] Henri Cohen, Advanced topics in computational number 
 71  .. [Cohen2000] Henri Cohen, Advanced topics in computational number 
72  72  theory, Graduate Texts in Mathematics, vol. 193, 
73  73  SpringerVerlag, New York, 2000. 
74  74  
75   .. [M] Jacques Martinet, Petits discriminants des corps de nombres, Journ. Arithm. 1980, 
 75  .. [Martinet1980] Jacques Martinet, Petits discriminants des corps de nombres, Journ. Arithm. 1980, 
76  76  Cambridge Univ. Press, 1982, 151193. 
77  77  
78   .. [T] Kisao Takeuchi, Totally real algebraic number fields of 
 78  .. [Takeuchi1999] Kisao Takeuchi, Totally real algebraic number fields of 
79  79  degree 9 with small discriminant, Saitama Math. J. 
80  80  17 (1999), 6385 (2000). 
81  81  
82   .. [V] John Voight, Enumeration of totally real number fields 
83   of bounded root discriminant, to appear in 
84   Lect. Notes in Comp. Sci. 
 82  .. [Voight2008] John Voight, Enumeration of totally real number fields of bounded root 
 83  discriminant, Lect. Notes in Comp. Sci. 5011 (2008). 
85  84  
86  85  Authors 
87  86   
… 
… 

165  164   John Voight (20070903) 
166  165  
167  166  NOTES: 
168   The values are calculated by Martinet [M]_. 
 167  The values are calculated by Martinet [Martinet1980]_. 
169  168  """ 
170  169  
171  170  if n <= 10: 