# HG changeset patch
# User Andrey Novoseltsev <novoselt@gmail.com>
# Date 1372975788 21600
# Node ID f2ba3bc3c4773ef332b2140375e9a9b5943fd499
# Parent 4d62717de85ef513af7b7f9b7d50e1eabf841084
Fixes for typos.
diff git a/sage/geometry/polyhedron/lattice_euclidean_group_element.py b/sage/geometry/polyhedron/lattice_euclidean_group_element.py
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137  137  
138  138  def domain_dim(self): 
139  139  """ 
140   Return the rank of the domain lattice 
 140  Return the dimension of the domain lattice 
141  141  
142  142  EXAMPLES:: 
143  143  
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158  158  
159  159  def codomain_dim(self): 
160  160  """ 
161   Return the rank of the codomain lattice 
 161  Return the dimension of the codomain lattice 
162  162  
163  163  EXAMPLES:: 
164  164  
diff git a/sage/schemes/toric/weierstrass.py b/sage/schemes/toric/weierstrass.py
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361  361   ``polynomial``  a polynomial. The toric hypersurface 
362  362  equation. Can be either a cubic, a biquadric, or the 
363  363  hypersurface in `\mathbb{P}^2[1,1,2]`. The equation need not be 
364   in any standard form, only its Newton polynomial is used. 
 364  in any standard form, only its Newton polyhedron is used. 
365  365  
366  366   ``variables``  a list of variables of the parent polynomial 
367  367  ring or ``None`` (default). In the latter case, all variables 
368  368  are taken to be polynomial ring variables. If a subset of 
369   polynomial ring variables are given, the Weierstras form is 
 369  polynomial ring variables are given, the Weierstrass form is 
370  370  determined over the function field generated by the remaining 
371  371  variables. 
372  372  
373    ``transformation``  boolean (default: ``True``). Whether to 
 373   ``transformation``  boolean (default: ``False``). Whether to 
374  374  return the new variables that bring ``polynomial`` into 
375  375  Weierstrass form. 
376  376  
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386  386  
387  387  .. math:: 
388  388  
389   Y^2 = X^3 + f X Z^4 + Z^6 
 389  Y^2 = X^3 + f X Z^4 + g Z^6 
390  390  
391  391  when restricted to the toric hypersurface. 
392  392  
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432  432  (0, 27/4) 
433  433  
434  434  This allows you to work with either homogeneous or inhomogeneous 
435   variables. For exmple, here is the del Pezzo surface of degree 8:: 
 435  variables. For example, here is the del Pezzo surface of degree 8:: 
436  436  
437  437  sage: dP8 = toric_varieties.dP8() 
438  438  sage: dP8.inject_variables() 
diff git a/sage/schemes/toric/weierstrass_covering.py b/sage/schemes/toric/weierstrass_covering.py
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1  1  r""" 
2  2  Map to the Weierstrass form of a toric elliptic curve. 
3  3  
4   There are 16 reflexive polygons in 2d. Each defines a toric fano 
5   variety, which (in 2d) has a unique crepant resolution to a smooth 
6   toric surface. An anticanonical hypersurface defines a genusone curve 
 4  There are 16 reflexive polygons in 2d. Each defines a toric Fano 
 5  variety, which (since it is 2d) has a unique crepant resolution to a smooth 
 6  toric surface. An anticanonical hypersurface defines a genus one curve 
7  7  `C` in this ambient space, with Jacobian elliptic curve `J(C)` which 
8  8  can be defined by the Weierstrass model `y^2 = x^3 + f x + g`. The 
9   coefficients `f`, `g` can be computed with the 
 9  coefficients `f` and `g` can be computed with the 
10  10  :mod:`~sage.schemes.toric.weierstrass` module. The purpose of this 
11   model is to give an explicit rational map `C \to J(C)`. This is a 
 11  model is to give an explicit rational map `C \to J(C)`. This is an 
12  12  `n^2`cover, where `n` is the minimal multisection of `C`. 
13  13  
14  14  Since it is technically often easier to deal with polynomials than 
… 
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20  20  
21  21  .. math:: 
22  22  
23   Y^2 = X^3 + f X Z^4 + Z^6 
 23  Y^2 = X^3 + f X Z^4 + g Z^6 
24  24  
25  25  EXAMPLES:: 
26  26  
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30  30  (0, 27/4) 
31  31  
32  32  That is, this hypersurface `C \in \mathbb{P}^2` has a Weierstrass 
33   equation `Y^2 = X^3 + 0 \cdot X Z^4  \frac{27}{4} Z^6` where 
 33  equation `Y^2 = X^3 + 0 \cdot X Z^4  \frac{27}{4} Z^6` where 
34  34  `[X:Y:Z]` are projective coordinates on `\mathbb{P}^2[2,3,1]`. The 
35  35  form of the map `C\to J(C)` is:: 
36  36  
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71  71  
72  72  sage: P2_112 = toric_varieties.P2_112() 
73  73  sage: C = P2_112.anticanonical_hypersurface(coefficients=[1]*4); C 
74   Closed subscheme of 2d CPRFano toric variety covered by 3 affine patches defined by: 
 74  Closed subscheme of 2d CPRFano toric variety 
 75  covered by 3 affine patches defined by: 
75  76  z0^4 + z2^4 + z0*z1*z2 + z1^2 
76  77  sage: eq = C.defining_polynomials()[0] 
77  78  sage: f, g = WeierstrassForm(eq) 
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134  135   ``polynomial``  a polynomial. The toric hypersurface 
135  136  equation. Can be either a cubic, a biquadric, or the 
136  137  hypersurface in `\mathbb{P}^2[1,1,2]`. The equation need not be 
137   in any standard form, only its Newton polynomial is used. 
 138  in any standard form, only its Newton polyhedron is used. 
138  139  
139  140   ``variables``  a list of variables of the parent polynomial 
140  141  ring or ``None`` (default). In the latter case, all variables 
141  142  are taken to be polynomial ring variables. If a subset of 
142   polynomial ring variables are given, the Weierstras form is 
 143  polynomial ring variables are given, the Weierstrass form is 
143  144  determined over the function field generated by the remaining 
144  145  variables. 
145  146  
… 
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151  152  
152  153  .. math:: 
153  154  
154   Y^2 = X^3 + f X Z^4 + Z^6 
 155  Y^2 = X^3 + f X Z^4 + g Z^6 
155  156  
156  157  when restricted to the toric hypersurface. 
157  158  
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356  357  
357  358  Input/output is the same as :func:`WeierstrassMap`, except that 
358  359  the input polynomial must be a standard anticanonical hypersurface 
359   in weighted projective space `\mathbb{P}^1 \times \mathbb{P}^1`: 
 360  in the toric surface `\mathbb{P}^1 \times \mathbb{P}^1`: 
360  361  
361  362  EXAMPLES:: 
362  363  
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399  400  h = Q.h_covariant() 
400  401  if t is None: 
401  402  t = 1 
402   #return ( 4*g*(a00*s+V/2), 4*h, (a00*s+V/2)**3 ) 
403  403  return ( 4*g*t**2, 4*h*t**3, (a00*s+V/2) ) 
404  404  
405  405  