Ticket #4539: trac4539_fix_docs.patch
File trac4539_fix_docs.patch, 42.1 KB (added by , 5 years ago) 


sage/libs/singular/function.pyx
# HG changeset patch # User Simon King <simon.king@unijena.de> # Date 1317313679 7200 # Node ID d129c0622eb94d3871297ccb39de1edbf44eb729 # Parent 9f95d173ac536e34aa7c6f711843b4aa1632f166 #4539: Fix doc string format; fix doc test errors left over by the previous patches. diff git a/sage/libs/singular/function.pyx b/sage/libs/singular/function.pyx
a b 405 405 406 406 def is_singular_poly_wrapper(p): 407 407 """ 408 Checks if p is some data type corresponding to some singular ``poly`` `.409 408 Checks if p is some data type corresponding to some singular ``poly``. 409 410 410 EXAMPLE:: 411 411 412 sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural413 sage: from sage.matrix.constructor import Matrix414 412 sage: from sage.libs.singular.function import is_singular_poly_wrapper 415 sage: c=Matrix(2) 416 sage: c[0,1]=1 417 sage: P = NCPolynomialRing_plural(QQ, 2, 'x,y', c=c, d=Matrix(2)) 418 sage: (x,y)=P.gens() 413 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 414 sage: H.<x,y,z> = A.g_algebra({z*x:x*z+2*x, z*y:y*z2*y}) 419 415 sage: is_singular_poly_wrapper(x+y) 420 416 True 421 417 422 418 """ 423 419 return PY_TYPE_CHECK(p, MPolynomial_libsingular) or PY_TYPE_CHECK(p, NCPolynomial_plural) 424 420 … … 1640 1636 <Resolution> 1641 1637 sage: singular_list(resolution) 1642 1638 [[(2*y, 2, y + 1, 0), (0, 2, x  1, 0), (x*y  y, y + 1, 1, y), (x^2 + 1, x  1, 1, x)], [(x  1, y  1, 2*x, 2*y)], [(0)]] 1643 sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural 1644 sage: from sage.matrix.constructor import Matrix 1645 sage: c=Matrix(2) 1646 sage: c[0,1]=1 1647 sage: P = NCPolynomialRing_plural(QQ, 2, 'x,y', c=c, d=Matrix(2)) 1648 sage: (x,y)=P.gens() 1649 sage: I= Sequence([x*y,x+y], check=False, immutable=True)#P.ideal(x*y,x+y) 1639 1640 sage: A.<x,y> = FreeAlgebra(QQ, 2) 1641 sage: P.<x,y> = A.g_algebra({y*x:x*y}) 1642 sage: I= Sequence([x*y,x+y], check=False, immutable=True) 1650 1643 sage: twostd = singular_function("twostd") 1651 1644 sage: twostd(I) 1652 1645 [x + y, y^2] 
sage/libs/singular/groebner_strategy.pyx
diff git a/sage/libs/singular/groebner_strategy.pyx b/sage/libs/singular/groebner_strategy.pyx
a b 299 299 300 300 EXAMPLES:: 301 301 302 sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy303 sage: P.<x,y,z> = PolynomialRing(QQ)304 sage: I = Ideal([x+z,y+z+1])305 sage: strat = GroebnerStrategy(I); strat306 Groebner Strategy for ideal generated by 2 elements307 over Multivariate Polynomial Ring in x, y, z over Rational Field302 sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy 303 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 304 sage: H.<x,y,z> = A.g_algebra({y*x:x*yz, z*x:x*z+2*x, z*y:y*z2*y}) 305 sage: I = H.ideal([y^2, x^2, z^2H.one_element()]) 306 sage: NCGroebnerStrategy(I) 307 Groebner Strategy for ideal generated by 3 elements over Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, ncrelations: {y*x: x*y  z, z*y: y*z  2*y, z*x: x*z + 2*x} 308 308 309 309 TESTS:: 310 310 311 sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy 312 sage: strat = GroebnerStrategy(None) 311 sage: strat = NCGroebnerStrategy(None) 313 312 Traceback (most recent call last): 314 313 ... 315 TypeError: First parameter must be a multivariate polynomial ideal.314 TypeError: First parameter must be an ideal in a galgebra. 316 315 317 sage: P.<x,y,z> = PolynomialRing( QQ,order='neglex')316 sage: P.<x,y,z> = PolynomialRing(CC,order='neglex') 318 317 sage: I = Ideal([x+z,y+z+1]) 319 sage: strat = GroebnerStrategy(I)318 sage: strat = NCGroebnerStrategy(I) 320 319 Traceback (most recent call last): 321 320 ... 322 NotImplementedError: The local case is not implemented yet. 323 324 sage: P.<x,y,z> = PolynomialRing(CC,order='neglex') 325 sage: I = Ideal([x+z,y+z+1]) 326 sage: strat = GroebnerStrategy(I) 327 Traceback (most recent call last): 328 ... 329 TypeError: First parameter's ring must be multivariate polynomial ring via libsingular. 321 TypeError: First parameter must be an ideal in a galgebra. 330 322 331 sage: P.<x,y,z> = PolynomialRing(ZZ)332 sage: I = Ideal([x+z,y+z+1])333 sage: strat = GroebnerStrategy(I)334 Traceback (most recent call last):335 ...336 NotImplementedError: Only coefficient fields are implemented so far.337 338 323 """ 339 324 if not isinstance(L, NCPolynomialIdeal): 340 325 raise TypeError("First parameter must be an ideal in a galgebra.") 341 326 342 327 if not isinstance(L.ring(), NCPolynomialRing_plural): 343 raise TypeError("First parameter's ring must be multivariate polynomial ring via libsingular.")328 raise TypeError("First parameter's ring must be a galgebra.") 344 329 345 330 self._ideal = L 346 331 … … 381 366 """ 382 367 TEST:: 383 368 384 sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy 385 sage: P.<x,y,z> = PolynomialRing(GF(32003)) 386 sage: I = Ideal([x + z, y + z]) 387 sage: strat = GroebnerStrategy(I) 388 sage: del strat 369 sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy 370 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 371 sage: H.<x,y,z> = A.g_algebra({y*x:x*yz, z*x:x*z+2*x, z*y:y*z2*y}) 372 sage: I = H.ideal([y^2, x^2, z^2H.one_element()]) 373 sage: strat = NCGroebnerStrategy(I) 374 sage: del strat # indirect doctest 389 375 """ 390 376 cdef ring *oldRing = NULL 391 377 if self._strat: … … 412 398 """ 413 399 TEST:: 414 400 415 sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy 416 sage: P.<x,y,z> = PolynomialRing(GF(32003)) 417 sage: I = Ideal([x + z, y + z]) 418 sage: strat = GroebnerStrategy(I) 401 sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy 402 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 403 sage: H.<x,y,z> = A.g_algebra({y*x:x*yz, z*x:x*z+2*x, z*y:y*z2*y}) 404 sage: I = H.ideal([y^2, x^2, z^2H.one_element()]) 405 sage: strat = NCGroebnerStrategy(I) 419 406 sage: strat # indirect doctest 420 Groebner Strategy for ideal generated by 2 elements over 421 Multivariate Polynomial Ring in x, y, z over Finite Field of size 32003 407 Groebner Strategy for ideal generated by 3 elements over Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, ncrelations: {y*x: x*y  z, z*y: y*z  2*y, z*x: x*z + 2*x} 422 408 """ 423 409 return "Groebner Strategy for ideal generated by %d elements over %s"%(self._ideal.ngens(),self._parent) 424 410 … … 428 414 429 415 EXAMPLE:: 430 416 431 sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy 432 sage: P.<x,y,z> = PolynomialRing(GF(32003)) 433 sage: I = Ideal([x + z, y + z]) 434 sage: strat = GroebnerStrategy(I) 435 sage: strat.ideal() 436 Ideal (x + z, y + z) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 32003 417 sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy 418 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 419 sage: H.<x,y,z> = A.g_algebra({y*x:x*yz, z*x:x*z+2*x, z*y:y*z2*y}) 420 sage: I = H.ideal([y^2, x^2, z^2H.one_element()]) 421 sage: strat = NCGroebnerStrategy(I) 422 sage: strat.ideal() == I 423 True 424 437 425 """ 438 426 return self._ideal 439 427 … … 443 431 444 432 EXAMPLE:: 445 433 446 sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy 447 sage: P.<x,y,z> = PolynomialRing(GF(32003)) 448 sage: I = Ideal([x + z, y + z]) 449 sage: strat = GroebnerStrategy(I) 450 sage: strat.ring() 451 Multivariate Polynomial Ring in x, y, z over Finite Field of size 32003 434 sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy 435 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 436 sage: H.<x,y,z> = A.g_algebra({y*x:x*yz, z*x:x*z+2*x, z*y:y*z2*y}) 437 sage: I = H.ideal([y^2, x^2, z^2H.one_element()]) 438 sage: strat = NCGroebnerStrategy(I) 439 sage: strat.ring() is H 440 True 452 441 """ 453 442 return self._parent 454 443 … … 456 445 """ 457 446 EXAMPLE:: 458 447 459 sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy 460 sage: P.<x,y,z> = PolynomialRing(GF(19)) 461 sage: I = Ideal([P(0)]) 462 sage: strat = GroebnerStrategy(I) 463 sage: strat == GroebnerStrategy(I) 448 sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy 449 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 450 sage: H.<x,y,z> = A.g_algebra({y*x:x*yz, z*x:x*z+2*x, z*y:y*z2*y}) 451 sage: I = H.ideal([y^2, x^2, z^2H.one_element()]) 452 sage: strat = NCGroebnerStrategy(I) 453 sage: strat == NCGroebnerStrategy(I) 464 454 True 465 sage: I = Ideal([x+1,y+z])466 sage: strat == GroebnerStrategy(I)455 sage: I = H.ideal([y^2, x^2, z^2H.one_element()], side='twosided') 456 sage: strat == NCGroebnerStrategy(I) 467 457 False 468 458 """ 469 459 if not isinstance(other, NCGroebnerStrategy): 470 460 return cmp(type(self),other(type)) 471 461 else: 472 return cmp(self._ideal.gens(),(<NCGroebnerStrategy>other)._ideal.gens()) 462 return cmp((self._ideal.gens(),self._ideal.side()), 463 ((<NCGroebnerStrategy>other)._ideal.gens(), 464 (<NCGroebnerStrategy>other)._ideal.side())) 473 465 474 466 def __reduce__(self): 475 467 """ 476 468 EXAMPLE:: 477 469 478 sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy 479 sage: P.<x,y,z> = PolynomialRing(GF(32003)) 480 sage: I = Ideal([x + z, y + z]) 481 sage: strat = GroebnerStrategy(I) 470 sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy 471 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 472 sage: H.<x,y,z> = A.g_algebra({y*x:x*yz, z*x:x*z+2*x, z*y:y*z2*y}) 473 sage: I = H.ideal([y^2, x^2, z^2H.one_element()]) 474 sage: strat = NCGroebnerStrategy(I) 482 475 sage: loads(dumps(strat)) == strat 483 476 True 484 477 """ … … 491 484 492 485 EXAMPLE:: 493 486 494 sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy 495 sage: P.<x,y,z> = PolynomialRing(QQ) 496 sage: I = Ideal([x + z, y + z]) 497 sage: strat = GroebnerStrategy(I) 498 sage: strat.normal_form(x*y) # indirect doctest 499 z^2 500 sage: strat.normal_form(x + 1) 501 z + 1 487 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 488 sage: H.<x,y,z> = A.g_algebra({y*x:x*yz, z*x:x*z+2*x, z*y:y*z2*y}) 489 sage: JL = H.ideal([x^3, y^3, z^3  4*z]) 490 sage: JT = H.ideal([x^3, y^3, z^3  4*z], side='twosided') 491 sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy 492 sage: SL = NCGroebnerStrategy(JL.std()) 493 sage: ST = NCGroebnerStrategy(JT.std()) 494 sage: SL.normal_form(x*y^2) 495 x*y^2 496 sage: ST.normal_form(x*y^2) 497 y*z 502 498 503 TESTS::504 505 sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy506 sage: P.<x,y,z> = PolynomialRing(QQ)507 sage: I = Ideal([P(0)])508 sage: strat = GroebnerStrategy(I)509 sage: strat.normal_form(x)510 x511 sage: strat.normal_form(P(0))512 0513 499 """ 514 500 if unlikely(p._parent is not self._parent): 515 501 raise TypeError("parent(p) must be the same as this object's parent.") … … 526 512 """ 527 513 EXAMPLE:: 528 514 529 sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy 530 sage: P.<x,y,z> = PolynomialRing(GF(32003)) 531 sage: I = Ideal([x + z, y + z]) 532 sage: strat = GroebnerStrategy(I) 533 sage: loads(dumps(strat)) == strat # indirect doctest 515 sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy 516 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 517 sage: H.<x,y,z> = A.g_algebra({y*x:x*yz, z*x:x*z+2*x, z*y:y*z2*y}) 518 sage: I = H.ideal([y^2, x^2, z^2H.one_element()]) 519 sage: strat = NCGroebnerStrategy(I) 520 sage: loads(dumps(strat)) == strat # indirect doctest 534 521 True 535 522 """ 536 523 return NCGroebnerStrategy(I) 
sage/rings/ideal_monoid.py
diff git a/sage/rings/ideal_monoid.py b/sage/rings/ideal_monoid.py
a b 81 81 82 82 sage: R.<a> = QuadraticField(23) 83 83 sage: M = sage.rings.ideal_monoid.IdealMonoid(R) 84 sage: M(a) 84 sage: M(a) # indirect doctest 85 85 Fractional ideal (a) 86 86 sage: M([a4, 13]) 87 87 Fractional ideal (13, 1/2*a + 9/2) 88 88 """ 89 #print x, type(x) 90 if isinstance(x, ideal.Ideal_generic): 89 try: 90 side = x.side() 91 except (AttributeError,TypeError): 92 side = None 93 try: 91 94 x = x.gens() 92 y = self.__R.ideal(x) 95 except AttributeError: 96 pass 97 if side is None: 98 y = self.__R.ideal(x) 99 else: 100 y = self.__R.ideal(x,side=side) 93 101 y._set_parent(self) 94 102 return y 95 103 
sage/rings/polynomial/multi_polynomial_ideal.py
diff git a/sage/rings/polynomial/multi_polynomial_ideal.py b/sage/rings/polynomial/multi_polynomial_ideal.py
a b 2957 2957 Apparently, ``x*y^2y*z`` should be in the twosided, but not 2958 2958 in the left ideal:: 2959 2959 2960 sage: x*y^2y*z in JL 2960 sage: x*y^2y*z in JL #indirect doctest 2961 2961 False 2962 2962 sage: x*y^2y*z in JT 2963 2963 True 
sage/rings/polynomial/plural.pxd
diff git a/sage/rings/polynomial/plural.pxd b/sage/rings/polynomial/plural.pxd
a b 21 21 # cdef NCPolynomial_plural _one_element 22 22 # cdef NCPolynomial_plural _zero_element 23 23 24 cdef public object _relations 24 cdef public object _relations,_relations_commutative 25 25 pass 26 26 27 27 cdef class ExteriorAlgebra_plural(NCPolynomialRing_plural): 
sage/rings/polynomial/plural.pyx
diff git a/sage/rings/polynomial/plural.pyx b/sage/rings/polynomial/plural.pyx
a b 22 22 23 23  Alexander Dreyer (201007): noncommutative ring functionality and documentation 24 24 25  Simon King (201109): left and twosided ideals; normal forms; pickling; 26 documentation 27 25 28 The underlying libSINGULAR interface was implemented by 26 29 27  Martin Albrecht (200701): initial implementation28 29  Joel Mohler (200801): misc improvements, polishing30 31  Martin Albrecht (200808): added `\QQ(a)` and `\ZZ` support32 33  Simon King (200904): improved coercion34 35  Martin Albrecht (200905): added `\ZZ/n\ZZ` support, refactoring36 37  Martin Albrecht (200906): refactored the code to allow better38 reuse30  Martin Albrecht (200701): initial implementation 31 32  Joel Mohler (200801): misc improvements, polishing 33 34  Martin Albrecht (200808): added `\QQ(a)` and `\ZZ` support 35 36  Simon King (200904): improved coercion 37 38  Martin Albrecht (200905): added `\ZZ/n\ZZ` support, refactoring 39 40  Martin Albrecht (200906): refactored the code to allow better 41 reuse 39 42 40 43 TODO: 41 44 … … 46 49 We show how to construct various noncommutative polynomial rings:: 47 50 48 51 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 49 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 50 sage: P.inject_variables() 51 Defining x, y, z 52 sage: P.<x,y,z> = A.g_algebra(relations={y*x:x*y}, order = 'lex') 52 53 53 54 sage: P 54 55 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, ncrelations: {y*x: x*y} … … 57 58 x*y + 1/2 58 59 59 60 sage: A.<x,y,z> = FreeAlgebra(GF(17), 3) 60 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 61 sage: P.inject_variables() 62 Defining x, y, z 63 61 sage: P.<x,y,z> = A.g_algebra(relations={y*x:x*y}, order = 'lex') 64 62 sage: P 65 63 Noncommutative Multivariate Polynomial Ring in x, y, z over Finite Field of size 17, ncrelations: {y*x: x*y} 66 64 … … 68 66 x*y + 7 69 67 70 68 71 Raw use of this class:: 69 Raw use of this class; *this is not the intended use!* 70 :: 71 72 72 sage: from sage.matrix.constructor import Matrix 73 73 sage: c = Matrix(3) 74 74 sage: c[0,1] = 2 … … 77 77 78 78 sage: d = Matrix(3) 79 79 sage: d[0, 1] = 17 80 sage: P = QQ['x','y','z'] 81 sage: c = c.change_ring(P) 82 sage: d = d.change_ring(P) 80 83 81 84 sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural 82 sage: R.<x,y,z> = NCPolynomialRing_plural(QQ, 3, c = c, d = d, order='lex')85 sage: R.<x,y,z> = NCPolynomialRing_plural(QQ, c = c, d = d, order=TermOrder('lex',3),category=Algebras(QQ)) 83 86 sage: R 84 87 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, ncrelations: {y*x: 2*x*y + 17} 85 88 … … 90 93 sage: f = 57 * a^2*b + 43 * c + 1; f 91 94 57*x^2*y + 43*z + 1 92 95 93 sage: R.term_order()94 Lexicographic term order95 96 96 TESTS:: 97 97 98 98 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 99 99 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 100 sage: P.inject_variables() 101 Defining x, y, z 102 103 sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural, NCPolynomial_plural 104 sage: TestSuite(NCPolynomialRing_plural).run() 105 sage: TestSuite(NCPolynomial_plural).run() 100 sage: TestSuite(P).run() 101 sage: loads(dumps(P)) is P 102 True 103 106 104 """ 107 105 include "sage/ext/stdsage.pxi" 108 106 include "sage/ext/interrupt.pxi" … … 198 196 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 199 197 sage: H = A.g_algebra({y*x:x*yz, z*x:x*z+2*x, z*y:y*z2*y}) 200 198 sage: H is A.g_algebra({y*x:x*yz, z*x:x*z+2*x, z*y:y*z2*y}) # indirect doctest 199 True 201 200 202 201 """ 203 202 if names is None: … … 241 240 Category of algebras over Rational Field 242 241 sage: TestSuite(H).run() 243 242 243 Note that two variables commute if they are not part of the given 244 relations:: 245 246 sage: H.<x,y,z> = A.g_algebra({z*x:x*z+2*x, z*y:y*z2*y}) 247 sage: x*y == y*x 248 True 249 244 250 """ 245 251 def __init__(self, base_ring, names, c, d, order, category, check = True): 246 252 """ … … 258 264 259 265 ``self.gen(j)*self.gen(i) == c[i, j] * self.gen(i)*self.gen(j) + d[i, j],`` 260 266 261 where ``0 <= i < j < self.ngens()``. 267 where ``0 <= i < j < self.ngens()``. Note that two variables 268 commute if they are not part of one of these relations. 262 269  ``order``  term order 263 270  ``check``  check the noncommutative conditions (default: ``True``) 264 271 265 EXAMPLES:: 272 TESTS: 273 274 It is strongly recommended to construct a galgebra using 275 :class:`G_AlgFactory`. The following is just for documenting 276 the arguments of the ``__init__`` method:: 266 277 267 278 sage: from sage.matrix.constructor import Matrix 268 sage: c = Matrix(3) 269 sage: c[0,1] = 1 270 sage: c[0,2] = 1 271 sage: c[1,2] = 1 272 273 sage: d = Matrix(3) 274 sage: d[0, 1] = 17 279 sage: c0 = Matrix(3) 280 sage: c0[0,1] = 1 281 sage: c0[0,2] = 1 282 sage: c0[1,2] = 1 283 284 sage: d0 = Matrix(3) 285 sage: d0[0, 1] = 17 286 sage: P = QQ['x','y','z'] 287 sage: c = c0.change_ring(P) 288 sage: d = d0.change_ring(P) 275 289 276 290 sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural 277 sage: P.<x,y,z> = NCPolynomialRing_plural(QQ, c = c, d = d, order= 'lex')291 sage: P.<x,y,z> = NCPolynomialRing_plural(QQ, c = c, d = d, order=TermOrder('lex',3), category=Algebras(QQ)) 278 292 279 293 sage: P # indirect doctest 280 294 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, ncrelations: {y*x: x*y + 17} … … 289 303 Lexicographic term order 290 304 291 305 sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural 292 sage: P.<x,y,z> = NCPolynomialRing_plural(GF(7), 3, c = c, d = d, order='degrevlex') 306 sage: P = GF(7)['x','y','z'] 307 sage: c = c0.change_ring(P) 308 sage: d = d0.change_ring(P) 309 sage: P.<x,y,z> = NCPolynomialRing_plural(GF(7), c = c, d = d, order=TermOrder('degrevlex',3), category=Algebras(GF(7))) 293 310 294 311 sage: P # indirect doctest 295 312 Noncommutative Multivariate Polynomial Ring in x, y, z over Finite Field of size 7, ncrelations: {y*x: x*y + 3} … … 323 340 self.__ngens = n 324 341 self.__term_order = order 325 342 326 Ring.__init__(self, base_ring, names, category )343 Ring.__init__(self, base_ring, names, category=category) 327 344 self._populate_coercion_lists_() 328 345 329 346 #MPolynomialRing_generic.__init__(self, base_ring, n, names, order) … … 364 381 collection). 365 382 366 383 TESTS: 384 367 385 This example caused a segmentation fault with a previous version 368 of this method: 386 of this method:: 387 369 388 sage: import gc 370 389 sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural 371 390 sage: from sage.algebras.free_algebra import FreeAlgebra … … 394 413 Make sure element is a valid member of self, and return the constructed element. 395 414 396 415 EXAMPLES:: 416 397 417 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 398 399 418 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 400 419 401 420 We can construct elements from the base ring:: … … 403 422 sage: P(1/2) 404 423 1/2 405 424 406 407 425 and all kinds of integers:: 408 426 409 427 sage: P(17) 410 428 17 411 412 429 sage: P(int(19)) 413 430 19 414 415 431 sage: P(long(19)) 416 432 19 417 418 TESTS: :433 434 TESTS: 419 435 420 436 Check conversion from self:: 437 421 438 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 422 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 423 sage: P.inject_variables() 424 Defining x, y, z 439 sage: P.<x,y,z> = A.g_algebra(relations={y*x:x*y}, order = 'lex') 425 440 426 441 sage: P._element_constructor_(1/2) 427 442 1/2 … … 432 447 sage: P._element_constructor_(y*x) 433 448 x*y 434 449 435 Raw use of this class::436 sage: from sage.matrix.constructor import Matrix437 sage: c = Matrix(3)438 sage: c[0,1] = 2439 sage: c[0,2] = 1440 sage: c[1,2] = 1441 442 sage: d = Matrix(3)443 sage: d[0, 1] = 17444 445 sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural446 sage: R.<x,y,z> = NCPolynomialRing_plural(QQ, 3, c = c, d = d, order='lex')447 sage: R._element_constructor_(x*y)448 x*y449 450 sage: P._element_constructor_(17)451 17452 453 sage: P._element_constructor_(int(19))454 19455 456 450 Testing special cases:: 451 457 452 sage: P._element_constructor_(1) 458 453 1 459 454 … … 536 531 cpdef _coerce_map_from_(self, S): 537 532 """ 538 533 The only things that coerce into this ring are: 539  the integer ring 540  other localizations away from fewer primes 541 542 EXAMPLES:: 534 535  the integer ring 536  other localizations away from fewer primes 537 538 EXAMPLES:: 539 543 540 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 544 541 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 545 546 542 sage: P._coerce_map_from_(ZZ) 547 543 True 548 544 """ … … 553 549 554 550 555 551 def __hash__(self): 556 """ 557 Return a hash for this noncommutative ring, that is, a hash of the string 558 representation of this polynomial ring. 559 560 EXAMPLES:: 561 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 562 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 563 sage: hash(P) # somewhat random output 564 ... 565 566 TESTS:: 567 568 Check conversion from self:: 569 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 570 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 571 sage: from sage.matrix.constructor import Matrix 572 sage: c = Matrix(3) 573 sage: c[0,1] = 1 574 sage: c[0,2] = 1 575 sage: c[1,2] = 1 576 577 sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural 578 sage: R.<x,y,z> = NCPolynomialRing_plural(QQ, 3, c = c, d = Matrix(3), order='lex') 579 sage: hash(R) == hash(P) 580 True 581 """ 582 return hash(str(self.__repr__()) + str(self.term_order()) ) 583 584 585 def __cmp__(self, right): 586 r""" 587 Noncommutative polynomial rings are said to be equal if: 588 589  their base rings match, 590  their generator names match, 591  their term orderings match, and 592  their relations match. 552 """ 553 Return a hash for this noncommutative ring, that is, a hash of the string 554 representation of this polynomial ring. 555 556 NOTE: 557 558 Galgebras are unique parents, provided that the galgebra constructor 559 is used. Thus, the hash simply is the memory address of the galgebra 560 (so, it is a session hash, but no stable hash). It is possible to 561 destroy uniqueness of galgebras on purpose, but that's your own 562 problem if you do those things. 593 563 594 564 EXAMPLES:: 595 565 596 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 597 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 598 599 sage: P == P 600 True 601 sage: Q = A.g_algebra(relations={y*x:x*y}, order = 'lex') 602 sage: Q == P 603 True 604 605 sage: from sage.matrix.constructor import Matrix 606 sage: c = Matrix(3) 607 sage: c[0,1] = 1 608 sage: c[0,2] = 1 609 sage: c[1,2] = 1 610 sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural 611 sage: R.<x,y,z> = NCPolynomialRing_plural(QQ, 3, c = c, d = Matrix(3), order='lex') 612 sage: R == P 613 True 614 615 sage: c[0,1] = 2 616 sage: R.<x,y,z> = NCPolynomialRing_plural(QQ, 3, c = c, d = Matrix(3), order='lex') 617 sage: P == R 618 False 566 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 567 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 568 sage: {P:2}[P] # indirect doctest 569 2 570 619 571 """ 620 621 if PY_TYPE_CHECK(right, NCPolynomialRing_plural): 622 623 return cmp( (self.base_ring(), map(str, self.gens()), 624 self.term_order(), self._c, self._d), 625 (right.base_ring(), map(str, right.gens()), 626 right.term_order(), 627 (<NCPolynomialRing_plural>right)._c, 628 (<NCPolynomialRing_plural>right)._d) 629 ) 630 else: 631 return cmp(type(self),type(right)) 572 return id(self) 632 573 633 574 def __pow__(self, n, _): 634 575 """ 635 576 Return the free module of rank `n` over this ring. 636 577 578 NOTE: 579 580 This is not properly implemented yet. Thus, there is 581 a warning. 582 637 583 EXAMPLES:: 584 638 585 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 639 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 640 sage: P.inject_variables() 641 Defining x, y, z 642 643 sage: f = x^3 + y 644 sage: f^2 645 x^6 + y^2 586 sage: P.<x,y,z> = A.g_algebra(relations={y*x:x*y}, order = 'lex') 587 sage: P^3 588 d...: UserWarning: You are constructing a free module 589 over a noncommutative ring. Sage does not have a concept 590 of left/right and both sided modules, so be careful. 591 It's also not guaranteed that all multiplications are 592 done from the right side. 593 d...: UserWarning: You are constructing a free module 594 over a noncommutative ring. Sage does not have a concept 595 of left/right and both sided modules, so be careful. 596 It's also not guaranteed that all multiplications are 597 done from the right side. 598 Ambient free module of rank 3 over Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, ncrelations: {y*x: x*y} 599 646 600 """ 647 601 import sage.modules.all 648 602 return sage.modules.all.FreeModule(self, n) … … 653 607 654 608 EXAMPLES:: 655 609 656 sage: A.<x,y,z> = FreeAlgebra(QQ, 3)657 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex')658 sage: P.term_order()659 Lexicographic term order660 661 sage: P = A.g_algebra(relations={y*x:x*y})662 sage: P.term_order()663 Degree reverse lexicographic term order610 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 611 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 612 sage: P.term_order() 613 Lexicographic term order 614 615 sage: P = A.g_algebra(relations={y*x:x*y}) 616 sage: P.term_order() 617 Degree reverse lexicographic term order 664 618 """ 665 619 return self.__term_order 666 620 … … 668 622 """ 669 623 Return False. 670 624 625 .. todo:: Provide a mathematically correct answer. 626 671 627 EXAMPLES:: 672 628 673 sage: A.<x,y,z> = FreeAlgebra(QQ, 3)674 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex')675 sage: P.is_commutative()676 False629 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 630 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 631 sage: P.is_commutative() 632 False 677 633 """ 678 634 return False 679 635 … … 683 639 684 640 EXAMPLES:: 685 641 686 sage: A.<x,y,z> = FreeAlgebra(QQ, 3)687 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex')688 sage: P.is_field()689 False642 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 643 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 644 sage: P.is_field() 645 False 690 646 """ 691 647 return False 692 648 693 649 def _repr_(self): 694 650 """ 695 EXAMPLE: 696 sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural 697 sage: from sage.matrix.constructor import Matrix 698 sage: c=Matrix(2) 699 sage: c[0,1]=1 700 sage: P.<x,y> = NCPolynomialRing_plural(QQ, 2, c=c, d=Matrix(2)) 701 sage: P # indirect doctest 651 EXAMPLE:: 652 653 sage: A.<x,y> = FreeAlgebra(QQ, 2) 654 sage: H.<x,y> = A.g_algebra({y*x:x*y}) 655 sage: H # indirect doctest 702 656 Noncommutative Multivariate Polynomial Ring in x, y over Rational Field, ncrelations: {y*x: x*y} 703 657 sage: x*y 704 658 x*y … … 715 669 Return an internal list representation of the noncummutative ring. 716 670 717 671 EXAMPLES:: 718 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 719 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 720 sage: P._ringlist() 721 [0, ['x', 'y', 'z'], [['lp', (1, 1, 1)], ['C', (0,)]], [0], [ 0 1 1] 722 [ 0 0 1] 723 [ 0 0 0], [0 0 0] 724 [0 0 0] 725 [0 0 0]] 672 673 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 674 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 675 sage: P._ringlist() 676 [0, ['x', 'y', 'z'], [['lp', (1, 1, 1)], ['C', (0,)]], [0], [ 0 1 1] 677 [ 0 0 1] 678 [ 0 0 0], [0 0 0] 679 [0 0 0] 680 [0 0 0]] 726 681 """ 727 682 cdef ring* _ring = self._ring 728 683 if(_ring != currRing): rChangeCurrRing(_ring) 729 684 from sage.libs.singular.function import singular_function 730 685 ringlist = singular_function('ringlist') 731 686 result = ringlist(self, ring=self) 732 733 734 735 736 687 return result 737 688 738 689 739 690 def relations(self, add_commutative = False): 740 691 """ 692 Return the relations of this galgebra. 693 694 INPUT: 695 696 ``add_commutative`` (optional bool, default False) 697 698 OUTPUT: 699 700 The defining relations. There are some implicit relations: 701 Two generators commute if they are not part of any given 702 relation. The implicit relations are not provided, unless 703 ``add_commutative==True``. 704 741 705 EXAMPLE:: 742 706 743 sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural 744 sage: from sage.matrix.constructor import Matrix 745 sage: c=Matrix(2) 746 sage: c[0,1]=1 747 sage: P = NCPolynomialRing_plural(QQ, 2, 'x,y', c=c, d=Matrix(2)) 748 sage: P # indirect doctest 749 Noncommutative Multivariate Polynomial Ring in x, y over Rational Field, ncrelations: ... 707 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 708 sage: H.<x,y,z> = A.g_algebra({z*x:x*z+2*x, z*y:y*z2*y}) 709 sage: x*y == y*x 710 True 711 sage: H.relations() 712 {z*y: y*z  2*y, z*x: x*z + 2*x} 713 sage: H.relations(add_commutative=True) 714 {y*x: x*y, z*y: y*z  2*y, z*x: x*z + 2*x} 715 750 716 """ 717 if add_commutative: 718 if self._relations_commutative is not None: 719 return self._relations_commutative 720 721 from sage.algebras.free_algebra import FreeAlgebra 722 A = FreeAlgebra( self.base_ring(), self.ngens(), self.gens() ) 723 724 res = {} 725 n = self.ngens() 726 for r in range(0, n1, 1): 727 for c in range(r+1, n, 1): 728 res[ A.gen(c) * A.gen(r) ] = self.gen(c) * self.gen(r) # C[r, c] * P.gen(r) * P.gen(c) + D[r, c] 729 self._relations_commutative = res 730 return res 731 751 732 if self._relations is not None: 752 733 return self._relations 753 734 … … 758 739 n = self.ngens() 759 740 for r in range(0, n1, 1): 760 741 for c in range(r+1, n, 1): 761 if (self.gen(c) * self.gen(r) != self.gen(r) * self.gen(c)) or add_commutative:742 if (self.gen(c) * self.gen(r) != self.gen(r) * self.gen(c)): 762 743 res[ A.gen(c) * A.gen(r) ] = self.gen(c) * self.gen(r) # C[r, c] * P.gen(r) * P.gen(c) + D[r, c] 763 764 744 765 745 self._relations = res 766 746 return self._relations 767 747 … … 772 752 EXAMPLES:: 773 753 774 754 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 775 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 776 sage: P.inject_variables() 777 Defining x, y, z 778 755 sage: P.<x,y,z> = A.g_algebra(relations={y*x:x*y}, order = 'lex') 779 756 sage: P.ngens() 780 757 3 781 758 """ … … 797 774 sage: P.gen(),P.gen(1) 798 775 (x, y) 799 776 800 sage: P.gen(1) 801 y 777 Note that the generators are not cached:: 778 779 sage: P.gen(1) is P.gen(1) 780 False 781 802 782 """ 803 783 cdef poly *_p 804 784 cdef ring *_ring = self._ring … … 820 800 INPUT: 821 801 822 802  ``*gens``  list or tuple of generators (or several input arguments) 823 824 803  ``coerce``  bool (default: ``True``); this must be a 825 804 keyword argument. Only set it to ``False`` if you are certain 826 805 that each generator is already in the ring. 806  ``side``  string (either "left", which is the default, or "twosided") 807 Must be a keyword argument. Defines whether the ideal is a left ideal 808 or a twosided ideal. Right ideals are not implemented. 827 809 828 810 EXAMPLES:: 811 829 812 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 830 sage: P = A.g_algebra(relations={y*x:x*y}, order = 'lex') 831 sage: P.inject_variables() 832 Defining x, y, z 813 sage: P.<x,y,z> = A.g_algebra(relations={y*x:x*y}, order = 'lex') 833 814 834 815 sage: P.ideal([x + 2*y + 2*z1, 2*x*y + 2*y*zy, x^2 + 2*y^2 + 2*z^2x]) 835 Ideal (x + 2*y + 2*z  1, 2*x*y + 2*y*z  y, x^2  x + 2*y^2 + 2*z^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, ncrelations: {y*x: x*y} 816 Left Ideal (x + 2*y + 2*z  1, 2*x*y + 2*y*z  y, x^2  x + 2*y^2 + 2*z^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, ncrelations: {y*x: x*y} 817 sage: P.ideal([x + 2*y + 2*z1, 2*x*y + 2*y*zy, x^2 + 2*y^2 + 2*z^2x], side="twosided") 818 Twosided Ideal (x + 2*y + 2*z  1, 2*x*y + 2*y*z  y, x^2  x + 2*y^2 + 2*z^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, ncrelations: {y*x: x*y} 819 836 820 """ 837 # from sage.rings.polynomial.multi_polynomial_ideal import \838 # NCPolynomialIdeal839 821 coerce = kwds.get('coerce', True) 840 822 if len(gens) == 1: 841 823 gens = gens[0] … … 872 854 ring = singular_function('ring') 873 855 return ring(L, ring=self) 874 856 857 # TODO: Implement this properly! 875 858 # def quotient(self, I): 876 859 # """ 877 860 # Construct quotient ring of ``self`` and the twosided Groebner basis of `ideal` … … 943 926 2/3 944 927 945 928 TESTS:: 929 946 930 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 947 931 sage: R = A.g_algebra(relations={y*x:x*y}, order='lex') 948 932 sage: R.inject_variables() … … 1037 1021 False 1038 1022 1039 1023 TESTS:: 1024 1040 1025 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 1041 1026 sage: Q = A.g_algebra(relations={y*x:x*y}, order='lex') 1042 1027 sage: Q.inject_variables() … … 1164 1149 (y, 1/4*x*y + 2/7) 1165 1150 1166 1151 TESTS:: 1152 1167 1153 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 1168 1154 sage: Q = A.g_algebra(relations={y*x:x*y}, order='lex') 1169 1155 sage: Q.inject_variables() … … 1314 1300 return M 1315 1301 1316 1302 def unpickle_NCPolynomial_plural(NCPolynomialRing_plural R, d): 1303 """ 1304 Auxiliary function to unpickle a noncommutative polynomial. 1305 1306 TEST:: 1307 1308 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 1309 sage: H.<x,y,z> = A.g_algebra({y*x:x*yz, z*x:x*z+2*x, z*y:y*z2*y}) 1310 sage: p = x*y+2*z+4*x*y*z*x 1311 sage: loads(dumps(p)) == p # indirect doctest 1312 True 1313 1314 """ 1317 1315 cdef ring *r = R._ring 1318 1316 cdef poly *m, *p 1319 1317 cdef int _i, _e … … 1703 1701 1704 1702 sage: (x*z).reduce(I) 1705 1703 x 1706 sage: I.twostd() 1704 1705 The Groebner basis shows that the result is correct:: 1706 1707 sage: I.std() 1707 1708 Left Ideal (z^2  1, y*z  y, x*z + x, y^2, 2*x*y  z  1, x^2) of 1708 1709 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational 1709 1710 Field, ncrelations: {y*x: x*y  z, z*y: y*z  2*y, z*x: x*z + 2*x} … … 1833 1834 degree, which is the maximum degree of any monomial. 1834 1835 1835 1836 OUTPUT: 1836 integer 1837 1838 integer 1837 1839 1838 1840 EXAMPLES:: 1839 1841 … … 1970 1972  a monomial (very fast, but not as flexible) 1971 1973 1972 1974 OUTPUT: 1973 element of the parent of this element. 1975 1976 element of the parent of this element. 1974 1977 1975 1978 .. note:: 1976 1979 … … 2089 2092  ``mon``  a monomial 2090 2093 2091 2094 OUTPUT: 2092 coefficient in base ring 2095 2096 coefficient in base ring 2093 2097 2094 2098 SEE ALSO: 2095 For coefficients in a base ring of fewer variables, look at ``coefficient``. 2099 2100 For coefficients in a base ring of fewer variables, look at ``coefficient``. 2096 2101 2097 2102 EXAMPLES:: 2098 2103 … … 2212 2217 INPUT: 2213 2218 2214 2219  ``x``  a tuple or, in case of a singlevariable MPolynomial 2215 ring x can also be an integer.2220 ring x can also be an integer. 2216 2221 2217 2222 EXAMPLES:: 2218 2223 … … 2276 2281 INPUT: 2277 2282 2278 2283  ``as_ETuples``  (default: ``True``) if true returns the result as an list of ETuples 2279 2284 otherwise returns a list of tuples 2280 2285 2281 2286 2282 2287 EXAMPLES:: … … 2660 2665 2661 2666 EXAMPLES:: 2662 2667 2663 2668 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 2669 sage: H = A.g_algebra({z*x:x*z+2*x, z*y:y*z2*y}) 2670 sage: H.gen(2) # indirect doctest 2671 z 2672 2664 2673 """ 2665 2674 cdef NCPolynomial_plural p = PY_NEW(NCPolynomial_plural) 2666 2675 p._parent = <ParentWithBase>parent … … 2676 2685 Construct MPolynomialRing_libsingular from ringWrap, assumming the ground field to be base_ring 2677 2686 2678 2687 EXAMPLES:: 2688 2679 2689 sage: H.<x,y,z> = PolynomialRing(QQ, 3) 2680 2690 sage: from sage.libs.singular.function import singular_function 2681 2691 … … 2720 2730 Construct NCPolynomialRing_plural from ringWrap, assumming the ground field to be base_ring 2721 2731 2722 2732 EXAMPLES:: 2723 EXAMPLES::2724 2733 2725 2734 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) 2726 2735 sage: H = A.g_algebra({y*x:x*y1}) … … 2872 2881 if (r in alt_vars) and (c in alt_vars): 2873 2882 relations[ A.gen(c) * A.gen(r) ] =  A.gen(r) * A.gen(c) 2874 2883 2875 H = A.g_algebra(relations=relations, order=order)2884 cdef NCPolynomialRing_plural H = A.g_algebra(relations=relations, order=order) 2876 2885 I = H.ideal([H.gen(i) *H.gen(i) for i in alt_vars]).twostd() 2877 return H.quotient(I) 2886 L = H._ringlist() 2887 L[3] = I 2888 W = H._list_to_ring(L) 2889 return new_NRing(W, H.base_ring()) 2878 2890 2879 2891 cdef poly *addwithcarry(poly *tempvector, poly *maxvector, int pos, ring *_ring): 2880 2892 if p_GetExp(tempvector, pos, _ring) < p_GetExp(maxvector, pos, _ring):