NZMATH  1.2.0 About: NZMATH is a Python based number theory oriented calculation system. Fossies Dox: NZMATH-1.2.0.tar.gz  ("inofficial" and yet experimental doxygen-generated source code documentation) ring.py
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1 """
2 base classes for polynomial rings and rational function fields.
3 """
4
5 from __future__ import division
6 import nzmath.ring as ring
7 import nzmath.poly.termorder as termorder
8 import nzmath.poly.univar as univar
9 import nzmath.poly.multivar as multivar
10
11
13  """
14  The class of uni-/multivariate polynomial ring.
15  There's no need to specify the variable names.
16  """
17
18  _instances = {}
19
20  def __init__(self, coeffring, number_of_variables=1):
21  """
22  PolynomialRing(coeffring)
23  creates a polynomial ring for univariate polynomials, while
24  PolynomialRing(coeffring, n)
25  creates a polynomial ring for multivariate polynomials.
26  """
27  if not isinstance(coeffring, ring.Ring):
28  raise TypeError("%s should not be passed as ring" % coeffring.__class__)
29  ring.CommutativeRing.__init__(self)
30  self._coefficient_ring = coeffring
31  if number_of_variables == 1 and self._coefficient_ring.isfield():
32  self.properties.setIseuclidean(True)
33  if self._coefficient_ring.isufd():
34  self.properties.setIsufd(True)
36  self.properties.setIsnoetherian(True)
37  elif self._coefficient_ring.isdomain() in (True, False):
38  self.properties.setIsdomain(self._coefficient_ring.isdomain())
39  self.number_of_variables = number_of_variables
40
41  def getCoefficientRing(self):
42  """
43  Return the coefficient ring.
44  """
45  return self._coefficient_ring
46
47  def getQuotientField(self):
48  """
49  Return the quotient field of the ring if coefficient ring has
50  its quotient field. Otherwise, an exception will be raised.
51  """
52  coefficientField = self._coefficient_ring.getQuotientField()
53  return RationalFunctionField(coefficientField, self.number_of_variables)
54
55  def __eq__(self, other):
56  """
57  equality test
58  """
59  if self is other:
60  return True
61  if (isinstance(other, PolynomialRing) and
62  self._coefficient_ring == other._coefficient_ring and
63  self.number_of_variables == other.number_of_variables):
64  return True
65  return False
66
67  def __ne__(self, other):
68  """
69  not equal
70  """
71  return not self.__eq__(other)
72
73  def __hash__(self):
74  """
75  hash(self)
76  """
77  return (hash(self._coefficient_ring) ^ (self.number_of_variables * hash(self.__class__.__name__) + 1)) & 0x7fffffff
78
79  def __repr__(self):
80  """
81  Return 'PolynomialRing(Ring, #vars)'
82  """
83  if self.number_of_variables == 1:
84  return "%s(%s)" % (self.__class__.__name__, repr(self._coefficient_ring))
85  return "%s(%s, %d)" % (self.__class__.__name__, repr(self._coefficient_ring), self.number_of_variables)
86
87  def __str__(self):
88  """
89  Return R[][]
90  """
91  return str(self._coefficient_ring) + "[]" * self.number_of_variables
92
93  def __contains__(self, element):
94  """
95  `in' operator is provided for checking the element be in the
96  ring.
97  """
98  if element in self._coefficient_ring:
99  return True
100  elem_ring = ring.getRing(element)
101  if elem_ring is not None and elem_ring.issubring(self):
102  return True
103  return False
104
105  def issubring(self, other):
106  """
107  reports whether another ring contains this polynomial ring.
108  """
109  if self is other:
110  return True
111  elif isinstance(other, PolynomialRing):
112  if (self._coefficient_ring.issubring(other.getCoefficientRing()) and
113  self.number_of_variables <= other.number_of_variables):
114  return True
115  elif other.issubring(self._coefficient_ring):
116  return False
117  try:
118  return other.issuperring(self)
119  except RuntimeError:
120  # reach max recursion by calling each other
121  return False
122
123  def issuperring(self, other):
124  """
125  reports whether this polynomial ring contains another ring.
126  """
127  if self is other:
128  return True
129  elif self._coefficient_ring.issuperring(other):
130  return True
131  elif isinstance(other, PolynomialRing):
132  return (self._coefficient_ring.issuperring(other.getCoefficientRing()) and
133  self.number_of_variables >= other.number_of_variables)
134  try:
135  return other.issubring(self)
136  except RuntimeError:
137  # reach max recursion by calling each other
138  return False
139
140  def getCommonSuperring(self, other):
141  """
142  Return common superring of two rings.
143  """
144  if self.issuperring(other):
145  return self
146  elif other.issuperring(self):
147  return other
148  elif (not isinstance(other, PolynomialRing) and
149  other.issuperring(self._coefficient_ring)):
150  return self.__class__(other, self.number_of_variables)
151  try:
152  return other.getCommonSuperring(self)
153  except RuntimeError:
154  # reached recursion limit by calling on each other
155  raise TypeError("no common super ring")
156  except AttributeError:
157  # other doesn't have getCommonSuperring
158  raise TypeError("%s is not a ring" % str(other))
159
160  def getCharacteristic(self):
161  """
162  Return characteristic of the ring.
163  """
165
166  def createElement(self, seed):
167  """
168  Return an element in the ring made from seed.
169  """
170  if ring.getRing(seed) == self:
171  return seed
172  elif not seed:
173  return self._zero_polynomial()
174  elif seed in self._coefficient_ring:
175  return self._constant_polynomial(seed)
176  else:
177  return self._prepared_polynomial(seed)
178
179  def _zero_polynomial(self):
180  """
181  Return the zero polynomial in the polynomial ring.
182  """
183  if self.number_of_variables == 1:
184  import nzmath.poly.uniutil as uniutil
185  return uniutil.polynomial((), self._coefficient_ring)
186  else:
187  import nzmath.poly.multiutil as multiutil
188  return multiutil.polynomial((), coeffring=self._coefficient_ring, number_of_variables=self.number_of_variables)
189
190  def _constant_polynomial(self, seed):
191  """
192  Return a constant polynomial made from a constant seed.
193  seed should not be zero.
194  """
195  if self.number_of_variables == 1:
196  import nzmath.poly.uniutil as uniutil
197  return uniutil.polynomial({0: seed}, self._coefficient_ring)
198  else:
199  import nzmath.poly.multiutil as multiutil
200  const = (0,) * self.number_of_variables
201  return multiutil.polynomial({const: seed}, self._coefficient_ring)
202
203  def _prepared_polynomial(self, preparation):
204  """
205  Return a polynomial from given preparation, which is suited
206  for the first argument of uni-/multi-variable polynomials.
207  """
208  if self.number_of_variables == 1:
209  import nzmath.poly.uniutil as uniutil
210  return uniutil.polynomial(preparation, self._coefficient_ring)
211  else:
212  import nzmath.poly.multiutil as multiutil
213  return multiutil.polynomial(preparation, self._coefficient_ring)
214
215  def _get_one(self):
216  "getter for one"
217  if self._one is None:
219  return self._one
220
221  one = property(_get_one, None, None, "multiplicative unit")
222
223  def _get_zero(self):
224  "getter for zero"
225  if self._zero is None:
226  self._zero = self._zero_polynomial()
227  return self._zero
228
229  zero = property(_get_zero, None, None, "additive unit")
230
231  def gcd(self, a, b):
232  """
233  Return the greatest common divisor of given polynomials.
234  The polynomials must be in the polynomial ring.
235  If the coefficient ring is a field, the result is monic.
236  """
237  if hasattr(a, "gcd"):
238  return a.gcd(b)
239  elif hasattr(a, "subresultant_gcd"):
240  return a.subresultant_gcd(b)
241  raise NotImplementedError("no gcd")
242
243  def extgcd(self, a, b):
244  """
245  Return the tuple (u, v, d): d is the greatest common divisor
246  of given polynomials, and they satisfy d = u*a + v*b. The
247  polynomials must be in the polynomial ring. If the
248  coefficient ring is a field, the result is monic.
249  """
250  if hasattr(a, "extgcd"):
251  return a.extgcd(b)
252  raise NotImplementedError("no extgcd")
253
254  @classmethod
255  def getInstance(cls, coeffring, number_of_variables=1):
256  """
257  Return an instance of the class with specified coefficient ring
258  and number of variables.
259  """
260  if (coeffring, number_of_variables) not in cls._instances:
261  cls._instances[coeffring, number_of_variables] = cls(coeffring, number_of_variables)
262  return cls._instances[coeffring, number_of_variables]
263
264
266  """
267  A class to represent an ideal of univariate polynomial ring.
268  """
269  def __init__(self, generators, polyring):
270  """
271  Initialize an ideal in the given polyring (polynomial ring).
272
273  generators: a generator polynomial or a list of polynomials
274  """
275  if type(polyring) is not PolynomialRing:
276  raise TypeError("polyring has to be an instance of PolynomialRing")
277  ring.Ideal.__init__(self, generators, polyring)
278  self._normalize_generators()
279
280  def __contains__(self, elem):
281  """
282  Return whether elem is in the ideal or not.
283  """
284  if not elem.getRing().issubring(self.ring):
285  return False
286  if self.generators == [self.ring.zero]:
287  return elem == self.ring.zero
288  return not self.reduce(elem)
289
290  def __nonzero__(self):
291  """
292  Report whether the ideal is zero ideal or not. Of course,
293  False is for zero ideal.
294  """
295  return self.generators and self.generators != [self.ring.zero]
296
297  def __repr__(self):
298  """
299  Return repr string.
300  """
301  return "%s(%r, %r)" % (self.__class__.__name__, self.generators, self.ring)
302
303  def __str__(self):
304  """
305  Return str string.
306  """
307  return "(%s)%s" % (", ".join([termorder.ascending_order.format(g) for g in self.generators]), self.ring)
308
309  def issubset(self, other):
310  """
311  Report whether another ideal contains this ideal.
312  """
313  if self is other:
314  return True
315  for g in self.generators:
316  if g not in other:
317  return False
318  return True
319
320  def issuperset(self, other):
321  """
322  Report whether this ideal contains another ideal.
323  """
324  if self is other:
325  return True
326  for g in other.generators:
327  if g not in self:
328  return False
329  return True
330
331  def reduce(self, element):
332  """
333  Reduce the given element by the ideal. The result is an
334  element of the class which represents the equivalent class.
335  """
336  order = termorder.ascending_order
337  if isinstance(element, univar.PolynomialInterface):
338  reduced = element
339  else:
340  reduced = self.ring.createElement(element)
341  if not self:
342  if not reduced:
343  reduced = self.ring.zero
344  elif len(self.generators) == 1:
345  g = self.generators
346  if self.ring.iseuclidean():
347  if isinstance(g, univar.PolynomialInterface):
348  reduced %= g
349  else: # g is element of a field
350  reduced = self.ring.zero
351  elif self.ring.getCoefficientRing().iseuclidean():
352  # higher degree to lower degree
353  # subtract euclid quotient of lc * generator
354  if isinstance(g, univar.PolynomialInterface):
356  degree = order.degree(reduced)
357  for d in range(degree, g_degree - 1, -1):
358  q = reduced[d] // lc
359  reduced -= g.term_mul((d - g_degree, q))
360  else:
361  reduced = reduced.coefficients_map(lambda c: c % g)
362  elif self.ring.getCoefficientRing().iseuclidean():
363  # assert that the generators are sorted descending order
364  # of degrees.
365  for g in self.generators:
366  reduced = self._euclidean_reduce(reduced, g)
367  else:
368  raise NotImplementedError("should be implemented")
369  return reduced
370
371  def _euclidean_reduce(self, element, g):
372  """
373  Reduce an element of the ring by a polynomial g.
374  The coefficient ring has to be a Euclidean domain.
375  """
376  order = termorder.ascending_order
377  coeffring = self.ring.getCoefficientRing()
378  reduced = element
381  while degree >= g_degree:
382  if not (lc % g_lc):
383  reduced -= g.term_mul((degree - g_degree, lc // g_lc))
384  else:
385  u, v, common_divisor = coeffring.extgcd(lc, g_lc)
386  reduced = reduced.scalar_mul(u) + g.term_mul((degree - g_degree, v))
387  break
389  return reduced
390
392  """
393  Normalize generators
394  """
395  order = termorder.ascending_order
396  if len(self.generators) > 1 and self.ring.ispid():
397  g = self.generators
398  for t in self.generators:
399  g = self.ring.gcd(g, t)
400  self.generators = [g]
401  elif self.ring.getCoefficientRing().iseuclidean():
402  coeffring = self.ring.getCoefficientRing()
403  degree = order.degree
405  cand_stack = list(self.generators)
406  tentative = []
407  while cand_stack:
408  next = cand_stack.pop()
409  if not tentative:
410  tentative.append(next)
411  continue
412  next_degree = degree(next)
413  while tentative:
414  last = tentative.pop()
415  last_degree = degree(last)
416  if last_degree > next_degree:
417  cand_stack.append(last)
418  continue
419  next_lc, last_lc = lc(next), lc(last)
420  if last_degree == next_degree:
421  u, v, d = coeffring.extgcd(next_lc, last_lc)
422  # make new polynomial whose lc = d
423  head = next.scalar_mul(u) + last.scalar_mul(v)
424  next -= head.scalar_mul(next_lc // d)
425  last -= head.scalar_mul(last_lc // d)
426  assert degree(next) < next_degree
427  assert degree(last) < last_degree
429  if next:
430  cand_stack.append(next)
431  if last:
432  cand_stack.append(last)
433  break
434  elif not (next_lc % last_lc):
435  next -= last.scalar_mul(next_lc // last_lc)
436  cand_stack.append(next)
437  break
438  elif last_lc % next_lc:
439  u, v, d = coeffring.extgcd(next_lc, last_lc)
440  next = next.scalar_mul(u) + last.term_mul((next_degree - last_degree, v))
441  tentative.append(last)
442  tentative.append(next)
443  break
444  else:
445  tentative.append(next)
446  self.generators = tentative
447  else:
448  degree = order.degree
449  # sort in descending order
450  self.generators.sort(cmp=lambda a, b: cmp(degree(b), degree(a)))
451
452
454  """
455  The class for rational function fields.
456  """
457  _instances = {}
458
459  def __init__(self, field, number_of_variables):
460  """
461  RationalFunctionField(field, number_of_variables)
462
463  field: The field of coefficients.
464  number_of_variables: The number of variables.
465  """
466  if isinstance(field, ring.QuotientField):
467  basedomain = PolynomialRing(field.basedomain, number_of_variables)
468  else:
469  basedomain = PolynomialRing(field, number_of_variables)
470  ring.QuotientField.__init__(self, basedomain)
471  self.coefficient_field = field
472  self.number_of_variables = number_of_variables
473
474  def __repr__(self):
475  """
476  Return 'RationalFunctionField(Field, #vars)'
477  """
478  return "%s(%s, %d)" % (self.__class__.__name__, repr(self.coefficient_field), self.number_of_variables)
479
480  def __str__(self):
481  """
482  Return K()()
483  """
484  return str(self.coefficient_field) + "()" * self.number_of_variables
485
486  def __eq__(self, other):
487  """
488  equality test
489  """
490  if self is other:
491  return True
492  if not isinstance(other, RationalFunctionField):
493  return False
494  if (self.coefficient_field == other.coefficient_field and
495  self.number_of_variables == other.number_of_variables):
496  return True
497  elif isinstance(self.coefficient_field, RationalFunctionField):
498  return self.unnest() == other
499  elif isinstance(other.coefficient_field, RationalFunctionField):
500  return self == other.unnest()
501  return False
502
503  def __contains__(self, element):
504  """
505  Return True if an element is in the field.
506  """
507  if ring.getRing(element).issubring(self):
508  return True
509  return False
510
511  def __hash__(self):
512  """
513  Return hash corresponding to equality.
514  """
515  return hash(self.coefficient_field) + self.number_of_variables
516
517  def getQuotientField(self):
518  """
519  Return the quotient field (the field itself).
520  """
521  return self
522
524  """
525  Return the coefficient field.
526  This method is provided for common interface with PolynomialRing.
527  """
528  return self.coefficient_field
529
530  def issubring(self, other):
531  """
532  Return True if self is a subring of the other.
533  """
534  if self is other:
535  return True
536  elif (isinstance(other, RationalFunctionField) and
537  self.coefficient_field.issubring(other.coefficient_field) and
538  self.number_of_variables <= other.number_of_variables):
539  return True
540  return False
541
542  def issuperring(self, other):
543  """
544  Return True if self is a superring of the other.
545  """
546  if self is other:
547  return True
548  elif (isinstance(other, RationalFunctionField) and
549  self.coefficient_field.issuperring(other.coefficient_field) and
550  self.number_of_variables >= other.number_of_variables):
551  return True
552  elif self.coefficient_field.issuperring(other):
553  return True
554  else:
555  try:
556  if self.issuperring(other.getQuotientField()):
557  return True
558  except Exception:
559  return False
560  return False
561
562  def getCharacteristic(self):
563  """
564  The characteristic of a rational function field is same as of
565  its coefficient field.
566  """
568
569  def getCommonSuperring(self, other):
570  """
571  Return common superring.
572  """
573  if self.issuperring(other):
574  return self
575  elif other.issuperring(self):
576  return other
577  elif isinstance(other, (PolynomialRing, RationalFunctionField)):
578  return RationalFunctionField(self.coefficient_field.getCommonSuperring(other.getCoefficientRing()), max(self.number_of_variables, other.number_of_variables))
579  try:
580  return other.getCommonSuperring(self)
581  except RuntimeError:
582  # reached recursion limit by calling on each other
583  raise TypeError("no common super ring")
584  except AttributeError:
585  # other doesn't have getCommonSuperring
586  raise TypeError("%s is not a ring" % str(other))
587
588  def unnest(self):
589  """
590  if self is a nested RationalFunctionField i.e. its
591  coefficientField is also a RationalFunctionField, then the
592  function returns one level unnested RationalFunctionField.
593
594  For example:
595  RationalFunctionField(RationalFunctionField(Q, 1), 1).unnest()
596  returns
597  RationalFunctionField(Q, 2).
598  """
599  return RationalFunctionField(self.coefficient_field.coefficient_field, self.coefficient_field.number_of_variables + self.number_of_variables)
600
601  def createElement(self, *seedarg, **seedkwd):
602  """
603  Return an element of the field made from seed.
604  """
605  import nzmath.poly.ratfunc as ratfunc
606  return ratfunc.RationalFunction(*seedarg, **seedkwd)
607
608  def _get_one(self):
609  """
610  getter for one
611  """
612  if self._one is None:
613  import nzmath.poly.ratfunc as ratfunc
614  poly_one = self.basedomain.one
615  self._one = ratfunc.RationalFunction(poly_one, poly_one)
616  return self._one
617
618  one = property(_get_one, None, None, "multiplicative unit.")
619
620  def _get_zero(self):
621  "getter for zero"
622  if self._zero is None:
623  import nzmath.poly.ratfunc as ratfunc
624  poly_one = self.basedomain.one
625  poly_zero = self.basedomain.zero
626  self._zero = ratfunc.RationalFunction(poly_zero, poly_one)
627  return self._zero
628
629  zero = property(_get_zero, None, None, "additive unit.")
630
631  @classmethod
632  def getInstance(cls, coefffield, number_of_variables):
633  """
634  Return an instance of the class with specified coefficient ring
635  and number of variables.
636  """
637  if (coefffield, number_of_variables) not in cls._instances:
638  cls._instances[coefffield, number_of_variables] = cls(coefffield, number_of_variables)
639  return cls._instances[coefffield, number_of_variables]
nzmath.poly.ring.PolynomialRing.getCommonSuperring
def getCommonSuperring(self, other)
Definition: ring.py:140
nzmath.bigrange.range
def range(start, stop=None, step=None)
Definition: bigrange.py:19
nzmath.ring
Definition: ring.py:1
nzmath.poly.ring.PolynomialRing.gcd
def gcd(self, a, b)
Definition: ring.py:231
nzmath.poly.ring.PolynomialRing._zero_polynomial
def _zero_polynomial(self)
Definition: ring.py:179
nzmath.poly.multiutil
Definition: multiutil.py:1
nzmath.ring.CommutativeRing.isdomain
def isdomain(self)
Definition: ring.py:114
nzmath.poly.ring.RationalFunctionField.__eq__
def __eq__(self, other)
Definition: ring.py:486
nzmath.ring.QuotientField.basedomain
basedomain
Definition: ring.py:248
nzmath.poly.ring.PolynomialIdeal.issubset
def issubset(self, other)
Definition: ring.py:309
nzmath.poly.ring.PolynomialRing.__str__
def __str__(self)
Definition: ring.py:87
nzmath.poly.ring.PolynomialRing.number_of_variables
number_of_variables
Definition: ring.py:39
nzmath.poly.ring.RationalFunctionField._get_one
def _get_one(self)
Definition: ring.py:608
nzmath.poly.ring.PolynomialRing.__eq__
def __eq__(self, other)
Definition: ring.py:55
nzmath.poly.ring.PolynomialRing.createElement
def createElement(self, seed)
Definition: ring.py:166
nzmath.poly.termorder
Definition: termorder.py:1
nzmath.poly.univar
Definition: univar.py:1
nzmath.poly.ring.PolynomialRing
Definition: ring.py:12
nzmath.poly.ring.RationalFunctionField.__init__
def __init__(self, field, number_of_variables)
Definition: ring.py:459
nzmath.poly.ring.PolynomialRing._get_zero
def _get_zero(self)
Definition: ring.py:223
nzmath.ring.CommutativeRing.isnoetherian
def isnoetherian(self)
Definition: ring.py:121
nzmath.poly.ring.RationalFunctionField.createElement
def createElement(self, *seedarg, **seedkwd)
Definition: ring.py:601
nzmath.poly.ring.RationalFunctionField.__hash__
def __hash__(self)
Definition: ring.py:511
nzmath.ring.Ring
Definition: ring.py:8
nzmath.poly.ring.PolynomialRing._coefficient_ring
_coefficient_ring
Definition: ring.py:30
nzmath.poly.ring.PolynomialIdeal
Definition: ring.py:265
nzmath.poly.ring.PolynomialRing.getInstance
def getInstance(cls, coeffring, number_of_variables=1)
Definition: ring.py:255
nzmath.poly.ring.PolynomialRing.getCharacteristic
def getCharacteristic(self)
Definition: ring.py:160
nzmath.ring.CommutativeRing
Definition: ring.py:88
nzmath.poly.ring.RationalFunctionField.issuperring
def issuperring(self, other)
Definition: ring.py:542
nzmath.poly.ring.PolynomialIdeal.generators
generators
Definition: ring.py:286
nzmath.poly.ring.PolynomialRing.extgcd
def extgcd(self, a, b)
Definition: ring.py:243
nzmath.poly.ring.PolynomialRing.__contains__
def __contains__(self, element)
Definition: ring.py:93
nzmath.poly.ring.PolynomialRing.__ne__
def __ne__(self, other)
Definition: ring.py:67
nzmath.poly.ring.RationalFunctionField._instances
dictionary _instances
Definition: ring.py:457
nzmath.poly.ring.RationalFunctionField.coefficient_field
coefficient_field
Definition: ring.py:471
nzmath.poly.ring.PolynomialRing._one
_one
Definition: ring.py:218
nzmath.poly.ring.PolynomialIdeal.__init__
def __init__(self, generators, polyring)
Definition: ring.py:269
nzmath.poly.ring.RationalFunctionField.getCoefficientRing
def getCoefficientRing(self)
Definition: ring.py:523
nzmath.ring.CommutativeRing.isufd
def isufd(self)
Definition: ring.py:128
nzmath.poly.ring.PolynomialRing._zero
_zero
Definition: ring.py:226
nzmath.ring.QuotientField
Definition: ring.py:234
nzmath.gcd.gcd
def gcd(a, b)
Definition: gcd.py:7
nzmath.poly.ring.RationalFunctionField.issubring
def issubring(self, other)
Definition: ring.py:530
nzmath.poly.ratfunc
Definition: ratfunc.py:1
nzmath.poly.ring.RationalFunctionField.unnest
def unnest(self)
Definition: ring.py:588
nzmath.poly.ring.RationalFunctionField.getQuotientField
def getQuotientField(self)
Definition: ring.py:517
nzmath.poly.ring.PolynomialRing._constant_polynomial
def _constant_polynomial(self, seed)
Definition: ring.py:190
nzmath.poly.ring.PolynomialRing.issubring
def issubring(self, other)
Definition: ring.py:105
nzmath.compatibility.cmp
cmp
Definition: compatibility.py:20
nzmath.poly.ring.PolynomialIdeal.__str__
def __str__(self)
Definition: ring.py:303
nzmath.poly.uniutil
Definition: uniutil.py:1
nzmath.poly.ring.PolynomialIdeal.issuperset
def issuperset(self, other)
Definition: ring.py:320
nzmath.poly.multivar
Definition: multivar.py:1
nzmath.poly.ring.RationalFunctionField._zero
_zero
Definition: ring.py:626
nzmath.ring.Ideal.ring
ring
Definition: ring.py:476
nzmath.poly.ring.RationalFunctionField.getCharacteristic
def getCharacteristic(self)
Definition: ring.py:562
nzmath.poly.ring.PolynomialIdeal._euclidean_reduce
def _euclidean_reduce(self, element, g)
Definition: ring.py:371
nzmath.poly.ring.PolynomialRing._instances
dictionary _instances
Definition: ring.py:18
nzmath.poly.ring.RationalFunctionField._one
_one
Definition: ring.py:615
nzmath.poly.ring.PolynomialRing._prepared_polynomial
def _prepared_polynomial(self, preparation)
Definition: ring.py:203
nzmath.poly.ring.PolynomialRing.__repr__
def __repr__(self)
Definition: ring.py:79
nzmath.poly.ring.PolynomialRing.getQuotientField
def getQuotientField(self)
Definition: ring.py:47
nzmath.poly.ring.PolynomialRing.issuperring
def issuperring(self, other)
Definition: ring.py:123
nzmath.poly.ring.RationalFunctionField.__str__
def __str__(self)
Definition: ring.py:480
nzmath.poly.ring.RationalFunctionField.getCommonSuperring
def getCommonSuperring(self, other)
Definition: ring.py:569
nzmath.poly.ring.PolynomialIdeal.__contains__
def __contains__(self, elem)
Definition: ring.py:280
nzmath.ring.CommutativeRing.properties
properties
Definition: ring.py:103
nzmath.ring.Ideal
Definition: ring.py:460
nzmath.poly.ring.RationalFunctionField.__contains__
def __contains__(self, element)
Definition: ring.py:503
nzmath.poly.ring.PolynomialRing.__hash__
def __hash__(self)
Definition: ring.py:73
nzmath.poly.ring.PolynomialIdeal.__nonzero__
def __nonzero__(self)
Definition: ring.py:290
nzmath.poly.ring.PolynomialIdeal._normalize_generators
def _normalize_generators(self)
Definition: ring.py:391
nzmath.poly.ring.RationalFunctionField.number_of_variables
number_of_variables
Definition: ring.py:472
nzmath.poly.ring.PolynomialRing.getCoefficientRing
def getCoefficientRing(self)
Definition: ring.py:41
nzmath.poly.ring.RationalFunctionField.__repr__
def __repr__(self)
Definition: ring.py:474
nzmath.poly.ring.PolynomialIdeal.__repr__
def __repr__(self)
Definition: ring.py:297
nzmath.poly.ring.PolynomialRing.__init__
def __init__(self, coeffring, number_of_variables=1)
Definition: ring.py:20
nzmath.ring.CommutativeRing.isfield
def isfield(self)
Definition: ring.py:149
nzmath.poly.ring.RationalFunctionField._get_zero
def _get_zero(self)
Definition: ring.py:620
nzmath.poly.ring.PolynomialRing._get_one
def _get_one(self)
Definition: ring.py:215
nzmath.poly.ring.PolynomialIdeal.reduce
def reduce(self, element)
Definition: ring.py:331
nzmath.poly.ring.RationalFunctionField.getInstance
def getInstance(cls, coefffield, number_of_variables)
Definition: ring.py:632