Changeset 6606:cac6ec5f25c5
- Timestamp:
- 09/30/07 11:10:39 (6 years ago)
- Branch:
- default
- Location:
- sage
- Files:
-
- 35 edited
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calculus/calculus.py (modified) (1 diff)
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categories/pushout.py (modified) (1 diff)
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crypto/mq/sr.py (modified) (1 diff)
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ext/interactive_constructors_c.pyx (modified) (1 diff)
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matrix/matrix2.pyx (modified) (1 diff)
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misc/functional.py (modified) (1 diff)
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modules/free_module.py (modified) (2 diffs)
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rings/finite_field.py (modified) (1 diff)
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rings/fraction_field.py (modified) (4 diffs)
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rings/homset.py (modified) (1 diff)
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rings/morphism.py (modified) (6 diffs)
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rings/polynomial/groebner_fan.py (modified) (5 diffs)
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rings/polynomial/multi_polynomial.pyx (modified) (1 diff)
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rings/polynomial/multi_polynomial_element.py (modified) (7 diffs)
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rings/polynomial/multi_polynomial_ideal.py (modified) (17 diffs)
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rings/polynomial/multi_polynomial_libsingular.pyx (modified) (11 diffs)
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rings/polynomial/multi_polynomial_ring.py (modified) (5 diffs)
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rings/polynomial/multi_polynomial_ring_generic.pyx (modified) (1 diff)
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rings/polynomial/polynomial_ring.py (modified) (3 diffs)
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rings/polynomial/polynomial_ring_constructor.py (modified) (6 diffs)
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rings/polynomial/toy_buchberger.py (modified) (1 diff)
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rings/quotient_ring.py (modified) (3 diffs)
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rings/quotient_ring_element.py (modified) (1 diff)
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rings/ring.pyx (modified) (8 diffs)
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schemes/elliptic_curves/constructor.py (modified) (1 diff)
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schemes/generic/affine_space.py (modified) (3 diffs)
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schemes/generic/divisor.py (modified) (1 diff)
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schemes/generic/morphism.py (modified) (1 diff)
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schemes/generic/projective_space.py (modified) (6 diffs)
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schemes/generic/spec.py (modified) (2 diffs)
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schemes/plane_curves/constructor.py (modified) (1 diff)
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structure/coerce.pyx (modified) (2 diffs)
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structure/parent.pyx (modified) (1 diff)
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structure/parent_base.pyx (modified) (1 diff)
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structure/parent_gens.pyx (modified) (2 diffs)
Legend:
- Unmodified
- Added
- Removed
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sage/calculus/calculus.py
r6377 r6606 1044 1044 3*x^35 + 2*y^35 1045 1045 sage: parent(g) 1046 Polynomial Ring in x, y over Finite Field of size 71046 Multivariate Polynomial Ring in x, y over Finite Field of size 7 1047 1047 """ 1048 1048 vars = self.variables() -
sage/categories/pushout.py
r5875 r6606 317 317 TypeError: Ambiguous Base Extension 318 318 sage: pushout(ZZ['x,y,z'], QQ['w,x,z,t']) 319 Polynomial Ring in w, x, y, z, t over Rational Field319 Multivariate Polynomial Ring in w, x, y, z, t over Rational Field 320 320 321 321 Some other examples 322 322 sage: pushout(Zp(7)['y'], Frac(QQ['t'])['x,y,z']) 323 Polynomial Ring in x, y, z over Fraction Field of Univariate Polynomial Ring in t over 7-adic Field with capped relative precision 20323 Multivariate Polynomial Ring in x, y, z over Fraction Field of Univariate Polynomial Ring in t over 7-adic Field with capped relative precision 20 324 324 sage: pushout(ZZ['x,y,z'], Frac(ZZ['x'])['y']) 325 Polynomial Ring in y, z over Fraction Field of Univariate Polynomial Ring in x over Integer Ring325 Multivariate Polynomial Ring in y, z over Fraction Field of Univariate Polynomial Ring in x over Integer Ring 326 326 sage: pushout(MatrixSpace(RDF, 2, 2), Frac(ZZ['x'])) 327 327 Full MatrixSpace of 2 by 2 dense matrices over Fraction Field of Univariate Polynomial Ring in x over Real Double Field -
sage/crypto/mq/sr.py
r6461 r6606 22 22 23 23 sage: sr.R 24 Polynomial Ring in k100, k101, k102, k103, x100, x101, x102, x103, w100, w101, w102, w103, s000, s001, s002, s003, k000, k001, k002, k003 over Finite Field in a of size 2^424 Multivariate Polynomial Ring in k100, k101, k102, k103, x100, x101, x102, x103, w100, w101, w102, w103, s000, s001, s002, s003, k000, k001, k002, k003 over Finite Field in a of size 2^4 25 25 26 26 For SR(1,1,1,4) the ShiftRows matrix isn't that interresting: -
sage/ext/interactive_constructors_c.pyx
r5374 r6606 196 196 sage: S = quo(R, (x^3, x^2 + y^2), 'a,b') 197 197 sage: S 198 Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x^3, x^2 + y^2)198 Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^3, x^2 + y^2) 199 199 sage: a^2 200 200 -b^2 -
sage/matrix/matrix2.pyx
r6590 r6606 1149 1149 [x1 x0] 1150 1150 sage: M.kernel() 1151 Vector space of degree 2 and dimension 1 over Fraction Field of Polynomial Ring in x0, x1 over Rational Field1151 Vector space of degree 2 and dimension 1 over Fraction Field of Multivariate Polynomial Ring in x0, x1 over Rational Field 1152 1152 Basis matrix: 1153 1153 [ 1 -1] -
sage/misc/functional.py
r6510 r6606 714 714 sage: R, x = objgens(MPolynomialRing(QQ,3, 'x')) 715 715 sage: R 716 Polynomial Ring in x0, x1, x2 over Rational Field716 Multivariate Polynomial Ring in x0, x1, x2 over Rational Field 717 717 sage: x 718 718 (x0, x1, x2) -
sage/modules/free_module.py
r6564 r6606 58 58 sage: M = FreeModule(R,2) 59 59 sage: M.base_ring() 60 Polynomial Ring in x, y over Rational Field60 Multivariate Polynomial Ring in x, y over Rational Field 61 61 62 62 sage: VectorSpace(QQ, 10).base_ring() … … 669 669 sage: M = FreeModule(R,2) 670 670 sage: M.ambient_module() 671 Ambient free module of rank 2 over the integral domain Polynomial Ring in x, y over Rational Field671 Ambient free module of rank 2 over the integral domain Multivariate Polynomial Ring in x, y over Rational Field 672 672 673 673 sage: V = FreeModule(QQ, 4).span([[1,2,3,4], [1,0,0,0]]); V -
sage/rings/finite_field.py
r6518 r6606 545 545 Multivariate polynomials also coerce: 546 546 sage: R = k['x,y,z']; R 547 Polynomial Ring in x, y, z over Finite Field in a of size 5^2547 Multivariate Polynomial Ring in x, y, z over Finite Field in a of size 5^2 548 548 sage: k(R(2)) 549 549 2 -
sage/rings/fraction_field.py
r5479 r6606 85 85 Fraction Field of Univariate Polynomial Ring in x over Integer Ring 86 86 sage: FractionField(MPolynomialRing(RationalField(),2,'x')) 87 Fraction Field of Polynomial Ring in x0, x1 over Rational Field87 Fraction Field of Multivariate Polynomial Ring in x0, x1 over Rational Field 88 88 89 89 Dividing elements often implicitly creates elements of the fraction field. … … 182 182 sage: R = Frac(QQ['x,y']) 183 183 sage: R 184 Fraction Field of Polynomial Ring in x, y over Rational Field184 Fraction Field of Multivariate Polynomial Ring in x, y over Rational Field 185 185 sage: R.ring() 186 Polynomial Ring in x, y over Rational Field186 Multivariate Polynomial Ring in x, y over Rational Field 187 187 """ 188 188 return self.__R … … 259 259 EXAMPLES: 260 260 sage: R = Frac(PolynomialRing(QQ,'z',10)); R 261 Fraction Field of Polynomial Ring in z0, z1, z2, z3, z4, z5, z6, z7, z8, z9 over Rational Field261 Fraction Field of Multivariate Polynomial Ring in z0, z1, z2, z3, z4, z5, z6, z7, z8, z9 over Rational Field 262 262 sage: R.ngens() 263 263 10 … … 271 271 EXAMPLES: 272 272 sage: R = Frac(PolynomialRing(QQ,'z',10)); R 273 Fraction Field of Polynomial Ring in z0, z1, z2, z3, z4, z5, z6, z7, z8, z9 over Rational Field273 Fraction Field of Multivariate Polynomial Ring in z0, z1, z2, z3, z4, z5, z6, z7, z8, z9 over Rational Field 274 274 sage: R.0 275 275 z0 -
sage/rings/homset.py
r5476 r6606 93 93 sage: S.<a,b> = R.quotient(x^2 + y^2) 94 94 sage: phi = S.hom([b,a]); phi 95 Ring endomorphism of Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)95 Ring endomorphism of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2) 96 96 Defn: a |--> b 97 97 b |--> a -
sage/rings/morphism.py
r6530 r6606 82 82 sage: R.<x,y,z> = PolynomialRing(QQ,3) 83 83 sage: phi = R.hom([y,z,x^2]); phi 84 Ring endomorphism of Polynomial Ring in x, y, z over Rational Field84 Ring endomorphism of Multivariate Polynomial Ring in x, y, z over Rational Field 85 85 Defn: x |--> y 86 86 y |--> z … … 94 94 sage: phi = S.hom([a^2, -b]) 95 95 sage: phi 96 Ring endomorphism of Quotient of Polynomial Ring in x, y over Rational Field by the ideal (y^2 + 1)96 Ring endomorphism of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (y^2 + 1) 97 97 Defn: a |--> a^2 98 98 b |--> -b … … 287 287 sage: K = QQ # by the way :-) 288 288 sage: R.<a,b,c,d> = K[]; R 289 Polynomial Ring in a, b, c, d over Rational Field289 Multivariate Polynomial Ring in a, b, c, d over Rational Field 290 290 sage: S.<u> = K[]; S 291 291 Univariate Polynomial Ring in u over Rational Field 292 292 sage: f = R.hom([0,0,0,u], S); f 293 293 Ring morphism: 294 From: Polynomial Ring in a, b, c, d over Rational Field294 From: Multivariate Polynomial Ring in a, b, c, d over Rational Field 295 295 To: Univariate Polynomial Ring in u over Rational Field 296 296 Defn: a |--> 0 … … 365 365 sage: S.lift() 366 366 Set-theoretic ring morphism: 367 From: Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2, y)368 To: Polynomial Ring in x, y over Rational Field367 From: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2, y) 368 To: Multivariate Polynomial Ring in x, y over Rational Field 369 369 Defn: Choice of lifting map 370 370 sage: S.lift() == 0 … … 570 570 sage: phi = S.cover(); phi 571 571 Ring morphism: 572 From: Polynomial Ring in x, y over Rational Field573 To: Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)572 From: Multivariate Polynomial Ring in x, y over Rational Field 573 To: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2) 574 574 Defn: Natural quotient map 575 575 sage: phi(x+y) … … 624 624 sage: S.<a, b, c> = R.quo(x^3 + y^3 + z^3) 625 625 sage: phi = S.hom([b, c, a]); phi 626 Ring endomorphism of Quotient of Polynomial Ring in x, y, z over Rational Field by the ideal (x^3 + y^3 + z^3)626 Ring endomorphism of Quotient of Multivariate Polynomial Ring in x, y, z over Rational Field by the ideal (x^3 + y^3 + z^3) 627 627 Defn: a |--> b 628 628 b |--> c -
sage/rings/polynomial/groebner_fan.py
r4499 r6606 105 105 sage: G = I.groebner_fan(); G 106 106 Groebner fan of the ideal: 107 Ideal (x^2*y - z, -x + y^2*z, x*z^2 - y) of Polynomial Ring in x, y, z over Rational Field107 Ideal (x^2*y - z, -x + y^2*z, x*z^2 - y) of Multivariate Polynomial Ring in x, y, z over Rational Field 108 108 """ 109 109 self.__is_groebner_basis = is_groebner_basis … … 150 150 sage: G._gfan_maps() 151 151 (Ring morphism: 152 From: Polynomial Ring in x, y, z over Rational Field153 To: Polynomial Ring in a, b, c over Rational Field152 From: Multivariate Polynomial Ring in x, y, z over Rational Field 153 To: Multivariate Polynomial Ring in a, b, c over Rational Field 154 154 Defn: x |--> a 155 155 y |--> b 156 156 z |--> c, 157 157 Ring morphism: 158 From: Polynomial Ring in a, b, c over Rational Field159 To: Polynomial Ring in x, y, z over Rational Field158 From: Multivariate Polynomial Ring in a, b, c over Rational Field 159 To: Multivariate Polynomial Ring in x, y, z over Rational Field 160 160 Defn: a |--> x 161 161 b |--> y … … 259 259 [z^15 - z, y - z^11, x - z^9] 260 260 sage: X[0].ideal() 261 Ideal (z^15 - z, y - z^11, x - z^9) of Polynomial Ring in x, y, z over Rational Field261 Ideal (z^15 - z, y - z^11, x - z^9) of Multivariate Polynomial Ring in x, y, z over Rational Field 262 262 sage: X[:5] 263 263 [[z^15 - z, y - z^11, x - z^9], … … 528 528 sage: G 529 529 Groebner fan of the ideal: 530 Ideal (-3*x^2 + y^3, 2*x^2 - x - 2*y^3 - y + z^3) of Polynomial Ring in x, y, z over Rational Field530 Ideal (-3*x^2 + y^3, 2*x^2 - x - 2*y^3 - y + z^3) of Multivariate Polynomial Ring in x, y, z over Rational Field 531 531 sage: G.tropical_basis () 532 532 [-4*x^2 - x - y + z^3, -3*x^2 + y^3] … … 665 665 sage: G = R.ideal([x - z^3, y^2 - 13*x]).groebner_fan() 666 666 sage: G[0].ideal() 667 Ideal (-13*z^3 + y^2, -z^3 + x) of Polynomial Ring in x, y, z over Rational Field667 Ideal (-13*z^3 + y^2, -z^3 + x) of Multivariate Polynomial Ring in x, y, z over Rational Field 668 668 """ 669 669 return self.__groebner_fan.ring().ideal(self) -
sage/rings/polynomial/multi_polynomial.pyx
r5976 r6606 124 124 x^3 + (17*w^3 + 3*w)*x + w^5 + z^5 125 125 sage: parent(f.polynomial(x)) 126 Univariate Polynomial Ring in x over Polynomial Ring in w, z over Rational Field126 Univariate Polynomial Ring in x over Multivariate Polynomial Ring in w, z over Rational Field 127 127 128 128 sage: f.polynomial(w) -
sage/rings/polynomial/multi_polynomial_element.py
r6511 r6606 208 208 sage: f = (x + y)/3 209 209 sage: f.parent() 210 Polynomial Ring in x, y over Rational Field210 Multivariate Polynomial Ring in x, y over Rational Field 211 211 212 212 If we do the same over $\ZZ$ the result is the same as … … 216 216 sage: f = (x + y)/3 217 217 sage: f.parent() 218 Polynomial Ring in x, y over Rational Field218 Multivariate Polynomial Ring in x, y over Rational Field 219 219 sage: f = (x + y) * 1/3 220 220 sage: f.parent() 221 Polynomial Ring in x, y over Rational Field221 Multivariate Polynomial Ring in x, y over Rational Field 222 222 223 223 But we get a true fraction field if the denominator is not in … … 226 226 sage: f = x/y 227 227 sage: f.parent() 228 Fraction Field of Polynomial Ring in x, y over Integer Ring228 Fraction Field of Multivariate Polynomial Ring in x, y over Integer Ring 229 229 """ 230 230 return self.parent().fraction_field()(self.__element, right.__element) … … 467 467 468 468 sage: R.<x> = PolynomialRing(GF(7),1); R 469 Polynomial Ring in x over Finite Field of size 7469 Multivariate Polynomial Ring in x over Finite Field of size 7 470 470 sage: f = 5*x^2 + 3; f 471 471 -2*x^2 + 3 … … 509 509 2 510 510 sage: c.parent() 511 Polynomial Ring in x, y over Rational Field511 Multivariate Polynomial Ring in x, y over Rational Field 512 512 sage: c in MPolynomialRing(RationalField(), 2, names = ['x','y']) 513 513 True … … 524 524 sage: R.<x,y> = GF(389)[] 525 525 sage: parent(R(x*y+5).coefficient(R(1))) 526 Polynomial Ring in x, y over Finite Field of size 389526 Multivariate Polynomial Ring in x, y over Finite Field of size 389 527 527 """ 528 528 R = self.parent() … … 621 621 5*x*y^10 + x^2*z^9 + y*z^10 + z^11 622 622 sage: g.parent() 623 Polynomial Ring in x, y, z over Rational Field623 Multivariate Polynomial Ring in x, y, z over Rational Field 624 624 """ 625 625 if self.is_homogeneous(): -
sage/rings/polynomial/multi_polynomial_ideal.py
r6592 r6606 44 44 sage: S.<a,b> = R.quotient((x^2 + y^2, 17)) # optional -- requires Macaulay2 45 45 sage: S # optional 46 Quotient of Polynomial Ring in x, y over Integer Ring by the ideal (x^2 + y^2, 17)46 Quotient of Multivariate Polynomial Ring in x, y over Integer Ring by the ideal (x^2 + y^2, 17) 47 47 sage: a^2 + b^2 == 0 # optional 48 48 True … … 68 68 sage: K.<zeta> = CyclotomicField(3) 69 69 sage: R.<x,y,z> = K[]; R 70 Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 270 Multivariate Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 2 71 71 sage: i = ideal(x - zeta*y + 1, x^3 - zeta*y^3); i 72 Ideal (x + (-zeta)*y + 1, x^3 + (-zeta)*y^3) of Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 272 Ideal (x + (-zeta)*y + 1, x^3 + (-zeta)*y^3) of Multivariate Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 2 73 73 sage: i.groebner_basis() 74 74 [x + (-zeta)*y + 1, 3*y^3 + (6*zeta + 3)*y^2 + (3*zeta - 3)*y - zeta - 2] 75 75 sage: S = R.quotient(i); S 76 Quotient of Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 2 by the ideal (x + (-zeta)*y + 1, x^3 + (-zeta)*y^3)76 Quotient of Multivariate Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 2 by the ideal (x + (-zeta)*y + 1, x^3 + (-zeta)*y^3) 77 77 sage: S.0 - zeta*S.1 78 78 -1 … … 314 314 sage: I = (p*q^2, y-z^2)*R 315 315 sage: pd = I.complete_primary_decomposition(); pd 316 [(Ideal (z^2 + 1, y + 1) of Polynomial Ring in x, y, z over Rational Field, Ideal (z^2 + 1, y + 1) of Polynomial Ring in x, y, z over Rational Field), (Ideal (z^6 + 4*z^3 + 4, y - z^2) of Polynomial Ring in x, y, z over Rational Field, Ideal (z^3 + 2, y - z^2) ofPolynomial Ring in x, y, z over Rational Field)]316 [(Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field, Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field), (Ideal (z^6 + 4*z^3 + 4, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field, Ideal (z^3 + 2, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field)] 317 317 318 318 sage: I.complete_primary_decomposition(algorithm = 'gtz') 319 [(Ideal (z^2 + 1, y - z^2) of Polynomial Ring in x, y, z over Rational Field, Ideal (z^2 + 1, y - z^2) of Polynomial Ring in x, y, z over Rational Field), (Ideal (z^6 + 4*z^3 + 4, y - z^2) of Polynomial Ring in x, y, z over Rational Field, Ideal (z^3 + 2, y - z^2) ofPolynomial Ring in x, y, z over Rational Field)]319 [(Ideal (z^2 + 1, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field, Ideal (z^2 + 1, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field), (Ideal (z^6 + 4*z^3 + 4, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field, Ideal (z^3 + 2, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field)] 320 320 """ 321 321 try: … … 345 345 sage: I = (p*q^2, y-z^2)*R 346 346 sage: I.primary_decomposition() 347 [Ideal (z^2 + 1, y + 1) of Polynomial Ring in x, y, z over Rational Field, Ideal (z^6 + 4*z^3 + 4, y - z^2) ofPolynomial Ring in x, y, z over Rational Field]347 [Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field, Ideal (z^6 + 4*z^3 + 4, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field] 348 348 349 349 """ … … 357 357 sage: I = (p*q^2, y-z^2)*R 358 358 sage: I.associated_primes() 359 [Ideal (y + 1, z^2 + 1) of Polynomial Ring in x, y, z over Rational Field, Ideal (z^2 - y, y*z + 2, y^2 + 2*z) ofPolynomial Ring in x, y, z over Rational Field]359 [Ideal (y + 1, z^2 + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field, Ideal (z^2 - y, y*z + 2, y^2 + 2*z) of Multivariate Polynomial Ring in x, y, z over Rational Field] 360 360 """ 361 361 return [P for _,P in self.complete_primary_decomposition(algorithm)] … … 403 403 sage: R.<a,b,c,d> = PolynomialRing(QQ, 4, order='lex') 404 404 sage: I = sage.rings.ideal.Cyclic(R,4); I 405 Ideal (a + b + c + d, a*b + a*d + b*c + c*d, a*b*c + a*b*d + a*c*d + b*c*d, a*b*c*d - 1) of Polynomial Ring in a, b, c, d over Rational Field405 Ideal (a + b + c + d, a*b + a*d + b*c + c*d, a*b*c + a*b*d + a*c*d + b*c*d, a*b*c*d - 1) of Multivariate Polynomial Ring in a, b, c, d over Rational Field 406 406 sage: I._groebner_basis_using_singular() 407 407 [c^2*d^6 - c^2*d^2 - d^4 + 1, c^3*d^2 + c^2*d^3 - c - d, b*d^4 - b + d^5 - d, b*c - b*d^5 + c^2*d^4 + c*d - d^6 - d^2, b^2 + 2*b*d + d^2, a + b + c + d] … … 447 447 sage: R.<a,b,c,d> = PolynomialRing(QQ, 4, order='lex') 448 448 sage: I = sage.rings.ideal.Cyclic(R,4); I 449 Ideal (a + b + c + d, a*b + a*d + b*c + c*d, a*b*c + a*b*d + a*c*d + b*c*d, a*b*c*d - 1) of Polynomial Ring in a, b, c, d over Rational Field449 Ideal (a + b + c + d, a*b + a*d + b*c + c*d, a*b*c + a*b*d + a*c*d + b*c*d, a*b*c*d - 1) of Multivariate Polynomial Ring in a, b, c, d over Rational Field 450 450 sage: I._groebner_basis_using_libsingular() 451 451 [c^2*d^6 - c^2*d^2 - d^4 + 1, c^3*d^2 + c^2*d^3 - c - d, b*d^4 - b + d^5 - d, b*c - b*d^5 + c^2*d^4 + c*d - d^6 - d^2, b^2 + 2*b*d + d^2, a + b + c + d] … … 508 508 sage: J = y*R 509 509 sage: I.intersection(J) 510 Ideal (x*y) of Polynomial Ring in x, y over Rational Field510 Ideal (x*y) of Multivariate Polynomial Ring in x, y over Rational Field 511 511 512 512 The following simple example illustrates that the product need not equal the intersection. … … 514 514 sage: J = (y^2, x)*R 515 515 sage: K = I.intersection(J); K 516 Ideal (y^2, x*y, x^2) of Polynomial Ring in x, y over Rational Field516 Ideal (y^2, x*y, x^2) of Multivariate Polynomial Ring in x, y over Rational Field 517 517 sage: IJ = I*J; IJ 518 Ideal (x^2*y^2, x^3, y^3, x*y) of Polynomial Ring in x, y over Rational Field518 Ideal (x^2*y^2, x^3, y^3, x*y) of Multivariate Polynomial Ring in x, y over Rational Field 519 519 sage: IJ == K 520 520 False … … 540 540 sage: I = (p*q^2, y-z^2)*R 541 541 sage: I.minimal_associated_primes () 542 [Ideal (z^3 + 2, -z^2 + y) of Polynomial Ring in x, y, z over Rational Field, Ideal (z^2 + 1, -z^2 + y) ofPolynomial Ring in x, y, z over Rational Field]542 [Ideal (z^3 + 2, -z^2 + y) of Multivariate Polynomial Ring in x, y, z over Rational Field, Ideal (z^2 + 1, -z^2 + y) of Multivariate Polynomial Ring in x, y, z over Rational Field] 543 543 544 544 ALGORITHM: Uses Singular. … … 559 559 sage: I = (x^2, y^3, (x*z)^4 + y^3 + 10*x^2)*R 560 560 sage: I.radical() 561 Ideal (y, x) of Polynomial Ring in x, y, z over Rational Field561 Ideal (y, x) of Multivariate Polynomial Ring in x, y, z over Rational Field 562 562 563 563 That the radical is correct is clear from the Groebner basis. … … 569 569 sage: I = (p*q^2, y-z^2)*R 570 570 sage: I.radical() 571 Ideal (z^2 - y, y^2*z + y*z + 2*y + 2) of Polynomial Ring in x, y, z over Rational Field571 Ideal (z^2 - y, y^2*z + y*z + 2*y + 2) of Multivariate Polynomial Ring in x, y, z over Rational Field 572 572 573 573 \note{(From Singular manual) A combination of the algorithms … … 579 579 sage: I = (p*q^2, y - z^2)*R 580 580 sage: I.radical() 581 Ideal (z^2 - y, y^2*z + y*z + 2*y + 2) of Polynomial Ring in x, y, z over Finite Field of size 37581 Ideal (z^2 - y, y^2*z + y*z + 2*y + 2) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 37 582 582 """ 583 583 S = self.ring() … … 793 793 sage: J = Ideal(I.transformed_basis('fglm',S)) 794 794 sage: J 795 Ideal (y^4 + y^3, x*y^3 - y^3, x^2 + y^3, z^4 + y^3 - y) of Polynomial Ring in z, x, y over Rational Field795 Ideal (y^4 + y^3, x*y^3 - y^3, x^2 + y^3, z^4 + y^3 - y) of Multivariate Polynomial Ring in z, x, y over Rational Field 796 796 sage: # example from the Singular manual page of gwalk 797 797 sage: R.<z,y,x>=PolynomialRing(GF(32003),3,order='lex') … … 844 844 sage: I = R * [x-t,y-t^2,z-t^3,s-x+y^3] 845 845 sage: I.elimination_ideal([t,s]) 846 Ideal (y^2 - x*z, x*y - z, x^2 - y) of Polynomial Ring in x, y, t, s, z over Rational Field846 Ideal (y^2 - x*z, x*y - z, x^2 - y) of Multivariate Polynomial Ring in x, y, t, s, z over Rational Field 847 847 848 848 ALGORITHM: Uses SINGULAR … … 904 904 sage: I = ideal(x*y-z^2, y^2-w^2) # optional 905 905 sage: I # optional 906 Ideal (x*y - z^2, y^2 - w^2) of Polynomial Ring in x, y, z, w over Integer Ring906 Ideal (x*y - z^2, y^2 - w^2) of Multivariate Polynomial Ring in x, y, z, w over Integer Ring 907 907 """ 908 908 #def __init__(self, ring, gens, coerce=True): … … 1004 1004 sage: R.<x,y> = PolynomialRing(IntegerRing(), 2, order='lex') 1005 1005 sage: R.ideal([x, y]) 1006 Ideal (x, y) of Polynomial Ring in x, y over Integer Ring1006 Ideal (x, y) of Multivariate Polynomial Ring in x, y over Integer Ring 1007 1007 sage: R.<x0,x1> = GF(3)[] 1008 1008 sage: R.ideal([x0^2, x1^3]) 1009 Ideal (x0^2, x1^3) of Polynomial Ring in x0, x1 over Finite Field of size 31009 Ideal (x0^2, x1^3) of Multivariate Polynomial Ring in x0, x1 over Finite Field of size 3 1010 1010 """ 1011 1011 Ideal_generic.__init__(self, ring, gens, coerce=coerce) -
sage/rings/polynomial/multi_polynomial_libsingular.pyx
r6605 r6606 153 153 sage: P.<x,y,z> = QQ[] 154 154 sage: P 155 Polynomial Ring in x, y, z over Rational Field155 Multivariate Polynomial Ring in x, y, z over Rational Field 156 156 157 157 sage: f = 27/113 * x^2 + y*z + 1/2; f … … 163 163 sage: P = MPolynomialRing(GF(127),3,names='abc', order='lex') 164 164 sage: P 165 Polynomial Ring in a, b, c over Finite Field of size 127165 Multivariate Polynomial Ring in a, b, c over Finite Field of size 127 166 166 167 167 sage: a,b,c = P.gens() … … 564 564 sage: P.<x,y> = QQ[] 565 565 sage: P 566 Polynomial Ring in x, y over Rational Field566 Multivariate Polynomial Ring in x, y over Rational Field 567 567 568 568 """ 569 569 varstr = ", ".join([ rRingVar(i,self._ring) for i in range(self.__ngens) ]) 570 return " Polynomial Ring in %s over %s"%(varstr,self._base)570 return "Multivariate Polynomial Ring in %s over %s"%(varstr,self._base) 571 571 572 572 def ngens(self): … … 653 653 sage: P.<x,y,z> = QQ[] 654 654 sage: sage.rings.ideal.Katsura(P) 655 Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y + 2*y*z - y) of Polynomial Ring in x, y, z over Rational Field655 Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y + 2*y*z - y) of Multivariate Polynomial Ring in x, y, z over Rational Field 656 656 657 657 sage: P.ideal([x + 2*y + 2*z-1, 2*x*y + 2*y*z-y, x^2 + 2*y^2 + 2*z^2-x]) 658 Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 + 2*y^2 + 2*z^2 - x) of Polynomial Ring in x, y, z over Rational Field658 Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 + 2*y^2 + 2*z^2 - x) of Multivariate Polynomial Ring in x, y, z over Rational Field 659 659 660 660 """ … … 873 873 sage: P.<x,y,z> = QQ[] 874 874 sage: hash(P) 875 -6257278808099690586# 64-bit875 967902441410893180 # 64-bit 876 876 -1767675994 # 32-bit 877 877 """ … … 1624 1624 sage: f = (x + y)/3 1625 1625 sage: f.parent() 1626 Polynomial Ring in x, y over Rational Field1626 Multivariate Polynomial Ring in x, y over Rational Field 1627 1627 1628 1628 Note that / is still a constructor for elements of the … … 1637 1637 (x^3 + y)/x 1638 1638 sage: h.parent() 1639 Fraction Field of Polynomial Ring in x, y over Rational Field1639 Fraction Field of Multivariate Polynomial Ring in x, y over Rational Field 1640 1640 1641 1641 TESTS: … … 2110 2110 2111 2111 sage: R.<x> = MPolynomialRing(GF(7),1); R 2112 Polynomial Ring in x over Finite Field of size 72112 Multivariate Polynomial Ring in x over Finite Field of size 7 2113 2113 sage: f = 5*x^2 + 3; f 2114 2114 -2*x^2 + 3 … … 2176 2176 2 2177 2177 sage: c.parent() 2178 Polynomial Ring in x, y over Rational Field2178 Multivariate Polynomial Ring in x, y over Rational Field 2179 2179 sage: c in P 2180 2180 True … … 2202 2202 sage: R.<x,y> = MPolynomialRing(GF(389),2) 2203 2203 sage: parent(R(x*y+5).coefficient(R(1))) 2204 Polynomial Ring in x, y over Finite Field of size 3892204 Multivariate Polynomial Ring in x, y over Finite Field of size 389 2205 2205 """ 2206 2206 cdef poly *p = self._poly … … 2342 2342 5*x*y^10 + x^2*z^9 + y*z^10 + z^11 2343 2343 sage: g.parent() 2344 Polynomial Ring in x, y, z over Rational Field2344 Multivariate Polynomial Ring in x, y, z over Rational Field 2345 2345 sage: f.homogenize(x) 2346 2346 2*x^11 + x^10*y + 5*x*y^10 -
sage/rings/polynomial/multi_polynomial_ring.py
r6382 r6606 151 151 EXAMPLES: 152 152 sage: R = MPolynomialRing(Integers(12), 'x', 5); R 153 Polynomial Ring in x0, x1, x2, x3, x4 over Ring of integers modulo 12153 Multivariate Polynomial Ring in x0, x1, x2, x3, x4 over Ring of integers modulo 12 154 154 sage.: loads(R.dumps()) == R # TODO -- this currently hangs sometimes (??) 155 155 True … … 210 210 -4*y^3 + x^2 211 211 sage: parent(f) 212 Polynomial Ring in x, y over Rational Field212 Multivariate Polynomial Ring in x, y over Rational Field 213 213 sage: parent(S(f)) 214 Polynomial Ring in x, y over Integer Ring214 Multivariate Polynomial Ring in x, y over Integer Ring 215 215 216 216 We coerce from the finite field. … … 218 218 3*y^3 + x^2 219 219 sage: parent(f) 220 Polynomial Ring in x, y over Rational Field220 Multivariate Polynomial Ring in x, y over Rational Field 221 221 222 222 We dump and load a the polynomial ring S: … … 248 248 <type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'> 249 249 sage: parent(f) 250 Polynomial Ring in x, y, z over Rational Field250 Multivariate Polynomial Ring in x, y, z over Rational Field 251 251 252 252 A more complicated symbolic and computational mix. Behind the scenes … … 256 256 (-z^3 + y^3 + x^3)^10 257 257 sage: g = R(f); parent(g) 258 Polynomial Ring in x, y, z over Rational Field258 Multivariate Polynomial Ring in x, y, z over Rational Field 259 259 sage: (f - g).expand() 260 260 0 -
sage/rings/polynomial/multi_polynomial_ring_generic.pyx
r6592 r6606 173 173 174 174 def _repr_(self): 175 return " Polynomial Ring in %s over %s"%(", ".join(self.variable_names()), self.base_ring())175 return "Multivariate Polynomial Ring in %s over %s"%(", ".join(self.variable_names()), self.base_ring()) 176 176 177 177 def repr_long(self): -
sage/rings/polynomial/polynomial_ring.py
r6428 r6606 123 123 False 124 124 sage: R = PolynomialRing(ZZ,1,'w'); R 125 Polynomial Ring in w over Integer Ring125 Multivariate Polynomial Ring in w over Integer Ring 126 126 sage: is_PolynomialRing(R) 127 127 False … … 467 467 Univariate Polynomial Ring in x over Integer Ring 468 468 sage: R.extend_variables('y, z') 469 Polynomial Ring in x, y, z over Integer Ring469 Multivariate Polynomial Ring in x, y, z over Integer Ring 470 470 sage: R.extend_variables(('y', 'z')) 471 Polynomial Ring in x, y, z over Integer Ring471 Multivariate Polynomial Ring in x, y, z over Integer Ring 472 472 """ 473 473 if isinstance(added_names, str): … … 1062 1062 x^2 + 2*x*y + y^2 + 2*x*z + 2*y*z + z^2 1063 1063 sage: parent(x) 1064 Polynomial Ring in x, y, z over Rational Field1064 Multivariate Polynomial Ring in x, y, z over Rational Field 1065 1065 sage: t = polygens(QQ,['x','yz','abc']) 1066 1066 sage: t -
sage/rings/polynomial/polynomial_ring_constructor.py
r5371 r6606 84 84 EXAMPLES of VARIABLE NAME CONTEXT: 85 85 sage: R.<x,y> = PolynomialRing(QQ,2); R 86 Polynomial Ring in x, y over Rational Field86 Multivariate Polynomial Ring in x, y over Rational Field 87 87 sage: f = x^2 - 2*y^2 88 88 … … 158 158 2. PolynomialRing(base_ring, names, order='degrevlex'): 159 159 sage: R = PolynomialRing(QQ, 'a,b,c'); R 160 Polynomial Ring in a, b, c over Rational Field160 Multivariate Polynomial Ring in a, b, c over Rational Field 161 161 162 162 sage: S = PolynomialRing(QQ, ['a','b','c']); S 163 Polynomial Ring in a, b, c over Rational Field163 Multivariate Polynomial Ring in a, b, c over Rational Field 164 164 165 165 sage: T = PolynomialRing(QQ, ('a','b','c')); T 166 Polynomial Ring in a, b, c over Rational Field166 Multivariate Polynomial Ring in a, b, c over Rational Field 167 167 168 168 All three rings are identical. … … 172 172 There is a unique polynomial ring with each term order: 173 173 sage: R = PolynomialRing(QQ, 'x,y,z', order='degrevlex'); R 174 Polynomial Ring in x, y, z over Rational Field174 Multivariate Polynomial Ring in x, y, z over Rational Field 175 175 sage: S = PolynomialRing(QQ, 'x,y,z', order='revlex'); S 176 Polynomial Ring in x, y, z over Rational Field176 Multivariate Polynomial Ring in x, y, z over Rational Field 177 177 sage: S is PolynomialRing(QQ, 'x,y,z', order='revlex') 178 178 True … … 186 186 variables, then variables labeled with numbers are created. 187 187 sage: PolynomialRing(QQ, 'x', 10) 188 Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9 over Rational Field188 Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9 over Rational Field 189 189 190 190 sage: PolynomialRing(GF(7), 'y', 5) 191 Polynomial Ring in y0, y1, y2, y3, y4 over Finite Field of size 7191 Multivariate Polynomial Ring in y0, y1, y2, y3, y4 over Finite Field of size 7 192 192 193 193 sage: PolynomialRing(QQ, 'y', 3, sparse=True) 194 Polynomial Ring in y0, y1, y2 over Rational Field194 Multivariate Polynomial Ring in y0, y1, y2 over Rational Field 195 195 196 196 It is easy in Python to create fairly aribtrary variable names. … … 199 199 200 200 sage: R = PolynomialRing(ZZ, ['x%s'%p for p in primes(100)]); R 201 Polynomial Ring in x2, x3, x5, x7, x11, x13, x17, x19, x23, x29, x31, x37, x41, x43, x47, x53, x59, x61, x67, x71, x73, x79, x83, x89, x97 over Integer Ring201 Multivariate Polynomial Ring in x2, x3, x5, x7, x11, x13, x17, x19, x23, x29, x31, x37, x41, x43, x47, x53, x59, x61, x67, x71, x73, x79, x83, x89, x97 over Integer Ring 202 202 203 203 By calling the \code{inject_variables()} method all those variable … … 210 210 You can also call \code{injvar}, which is a convenient shortcut for \code{inject_variables()}. 211 211 sage: R = PolynomialRing(GF(7),15,'w'); R 212 Polynomial Ring in w0, w1, w2, w3, w4, w5, w6, w7, w8, w9, w10, w11, w12, w13, w14 over Finite Field of size 7212 Multivariate Polynomial Ring in w0, w1, w2, w3, w4, w5, w6, w7, w8, w9, w10, w11, w12, w13, w14 over Finite Field of size 7 213 213 sage: R.injvar() 214 214 Defining w0, w1, w2, w3, w4, w5, w6, w7, w8, w9, w10, w11, w12, w13, w14 -
sage/rings/polynomial/toy_buchberger.py
r5619 r6606 61 61 62 62 sage: I 63 Ideal (a + 2*b + 2*c - 1, a^2 + 2*b^2 + 2*c^2 - a, 2*a*b + 2*b*c - b) of Polynomial Ring in a, b, c over Finite Field of size 12763 Ideal (a + 2*b + 2*c - 1, a^2 + 2*b^2 + 2*c^2 - a, 2*a*b + 2*b*c - b) of Multivariate Polynomial Ring in a, b, c over Finite Field of size 127 64 64 65 65 The original Buchberger algorithm performs 15 useless reductions to zero for this example: -
sage/rings/quotient_ring.py
r6383 r6606 86 86 sage: T.<c,d> = S.quo(a) 87 87 sage: T 88 Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x, y^2 + 1)88 Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x, y^2 + 1) 89 89 sage: T.gens() 90 90 (0, d) … … 139 139 sage: l = pi.lift(); l 140 140 Set-theoretic ring morphism: 141 From: Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x^2, y^2)142 To: Polynomial Ring in x, y over Rational Field141 From: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2, y^2) 142 To: Multivariate Polynomial Ring in x, y over Rational Field 143 143 Defn: Choice of lifting map 144 144 sage: l(x+y^3) … … 164 164 sage: pi = S.cover(); pi 165 165 Ring morphism: 166 From: Polynomial Ring in x, y over Rational Field167 To: Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)166 From: Multivariate Polynomial Ring in x, y over Rational Field 167 To: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2) 168 168 Defn: Natural quotient map 169 169 sage: L = S.lift(); L 170 170 Set-theoretic ring morphism: 171 From: Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)172 To: Polynomial Ring in x, y over Rational Field171 From: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2) 172 To: Multivariate Polynomial Ring in x, y over Rational Field 173 173 Defn: Choice of lifting map 174 174 sage: L(S.0) -
sage/rings/quotient_ring_element.py
r4483 r6606 47 47 sage: R.<x,y> = PolynomialRing(QQ, 2) 48 48 sage: S = R.quo(x^2 + y^2); S 49 Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)49 Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2) 50 50 sage: S.gens() 51 51 (xbar, ybar) -
sage/rings/ring.pyx
r6375 r6606 60 60 Univariate Polynomial Ring in abc over Finite Field of size 17 61 61 sage: GF(17)['a,b,c'] 62 Polynomial Ring in a, b, c over Finite Field of size 1762 Multivariate Polynomial Ring in a, b, c over Finite Field of size 17 63 63 64 64 We can also create power series rings (in one variable) by … … 165 165 sage: R.<x,y> = QQ[] 166 166 sage: R.ideal(x,y) 167 Ideal (x, y) of Polynomial Ring in x, y over Rational Field167 Ideal (x, y) of Multivariate Polynomial Ring in x, y over Rational Field 168 168 sage: R.ideal(x+y^2) 169 Ideal (y^2 + x) of Polynomial Ring in x, y over Rational Field169 Ideal (y^2 + x) of Multivariate Polynomial Ring in x, y over Rational Field 170 170 sage: R.ideal( [x^3,y^3+x^3] ) 171 Ideal (x^3, x^3 + y^3) of Polynomial Ring in x, y over Rational Field171 Ideal (x^3, x^3 + y^3) of Multivariate Polynomial Ring in x, y over Rational Field 172 172 """ 173 173 C = self._ideal_class_() … … 184 184 sage: R.<x,y,z> = GF(7)[] 185 185 sage: (x+y)*R 186 Ideal (x + y) of Polynomial Ring in x, y, z over Finite Field of size 7186 Ideal (x + y) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 7 187 187 sage: (x+y,z+y^3)*R 188 Ideal (x + y, y^3 + z) of Polynomial Ring in x, y, z over Finite Field of size 7188 Ideal (x + y, y^3 + z) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 7 189 189 """ 190 190 if isinstance(self, Ring): … … 207 207 sage: R.<x,y> = ZZ[] 208 208 sage: R.principal_ideal(x+2*y) 209 Ideal (x + 2*y) of Polynomial Ring in x, y over Integer Ring209 Ideal (x + 2*y) of Multivariate Polynomial Ring in x, y over Integer Ring 210 210 """ 211 211 return self.ideal([gen], coerce=coerce) … … 602 602 sage: R = Integers(389)['x,y'] 603 603 sage: Frac(R) 604 Fraction Field of Polynomial Ring in x, y over Ring of integers modulo 389604 Fraction Field of Multivariate Polynomial Ring in x, y over Ring of integers modulo 389 605 605 sage: R.fraction_field() 606 Fraction Field of Polynomial Ring in x, y over Ring of integers modulo 389606 Fraction Field of Multivariate Polynomial Ring in x, y over Ring of integers modulo 389 607 607 """ 608 608 if self.is_field(): … … 696 696 sage: S.<a,b> = R.quotient((x^2, y)) 697 697 sage: S 698 Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x^2, y)698 Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2, y) 699 699 sage: S.gens() 700 700 (a, 0) … … 719 719 sage: S.<a,b> = R.quo((x^2, y)) 720 720 sage: S 721 Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x^2, y)721 Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2, y) 722 722 sage: S.gens() 723 723 (a, 0) … … 949 949 Rational Field 950 950 sage: Frac(ZZ['x,y']).integral_closure() 951 Fraction Field of Polynomial Ring in x, y over Integer Ring951 Fraction Field of Multivariate Polynomial Ring in x, y over Integer Ring 952 952 """ 953 953 return self -
sage/schemes/elliptic_curves/constructor.py
r5772 r6606 81 81 sage: R = ZZ['u', 'v'] 82 82 sage: EllipticCurve(R, [1,1]) 83 Elliptic Curve defined by y^2 = x^3 + x +1 over Polynomial Ring in u, v83 Elliptic Curve defined by y^2 = x^3 + x +1 over Multivariate Polynomial Ring in u, v 84 84 over Integer Ring 85 85 """ -
sage/schemes/generic/affine_space.py
r3344 r6606 64 64 Affine Space of dimension 2 over Rational Field 65 65 sage: A.coordinate_ring() 66 Polynomial Ring in X, Y over Rational Field66 Multivariate Polynomial Ring in X, Y over Rational Field 67 67 68 68 Use the divide operator for base extension. … … 253 253 EXAMPLES: 254 254 sage: R = AffineSpace(2, GF(9,'alpha'), 'z').coordinate_ring(); R 255 Polynomial Ring in z0, z1 over Finite Field in alpha of size 3^2255 Multivariate Polynomial Ring in z0, z1 over Finite Field in alpha of size 3^2 256 256 sage: AffineSpace(3, R, 'x').coordinate_ring() 257 Polynomial Ring in x0, x1, x2 overPolynomial Ring in z0, z1 over Finite Field in alpha of size 3^2257 Multivariate Polynomial Ring in x0, x1, x2 over Multivariate Polynomial Ring in z0, z1 over Finite Field in alpha of size 3^2 258 258 """ 259 259 try: … … 345 345 (x, y^2, x*y^2) 346 346 sage: I = X.defining_ideal(); I 347 Ideal (x, y^2, x*y^2) of Polynomial Ring in x, y over Rational Field347 Ideal (x, y^2, x*y^2) of Multivariate Polynomial Ring in x, y over Rational Field 348 348 sage: I.groebner_basis() 349 349 [x, y^2] -
sage/schemes/generic/divisor.py
r4487 r6606 22 22 3 23 23 sage: D[1][1] 24 Ideal (x, y) of Polynomial Ring in x, y, z over Finite Field of size 524 Ideal (x, y) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 5 25 25 sage: C.divisor([(3, pts[0]), (-1, pts[1]), (10,pts[5])]) 26 26 10*(x + 2*z, y + z) + 3*(x, y) - (x, z) -
sage/schemes/generic/morphism.py
r6221 r6606 118 118 X: Spectrum of Univariate Polynomial Ring in x over Rational Field 119 119 Y: Spectrum of Univariate Polynomial Ring in y over Rational Field 120 U: Spectrum of Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x*y - 1)120 U: Spectrum of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x*y - 1) 121 121 122 122 sage: a, b = P1.gluing_maps() 123 123 sage: a 124 124 Affine Scheme morphism: 125 From: Spectrum of Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x*y - 1)125 From: Spectrum of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x*y - 1) 126 126 To: Spectrum of Univariate Polynomial Ring in x over Rational Field 127 127 Defn: Ring morphism: 128 128 From: Univariate Polynomial Ring in x over Rational Field 129 To: Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x*y - 1)129 To: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x*y - 1) 130 130 Defn: x |--> xbar 131 131 sage: b 132 132 Affine Scheme morphism: 133 From: Spectrum of Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x*y - 1)133 From: Spectrum of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x*y - 1) 134 134 To: Spectrum of Univariate Polynomial Ring in y over Rational Field 135 135 Defn: Ring morphism: 136 136 From: Univariate Polynomial Ring in y over Rational Field 137 To: Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x*y - 1)137 To: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x*y - 1) 138 138 Defn: y |--> ybar 139 139 """ -
sage/schemes/generic/projective_space.py
r4482 r6606 29 29 Projective Space of dimension 2 over Rational Field 30 30 sage: x.parent() 31 Polynomial Ring in x, y, z over Rational Field31 Multivariate Polynomial Ring in x, y, z over Rational Field 32 32 33 33 For example, we use $x,y,z$ to define the intersection of two lines. … … 86 86 Projective Space of dimension 2 over Rational Field 87 87 sage: P.coordinate_ring() 88 Polynomial Ring in X, Y, Z over Rational Field88 Multivariate Polynomial Ring in X, Y, Z over Rational Field 89 89 90 90 The divide operator does base extension. … … 101 101 Projective Space of dimension 2 over Finite Field of size 7 102 102 sage: P.coordinate_ring() 103 Polynomial Ring in x, y, z over Finite Field of size 7103 Multivariate Polynomial Ring in x, y, z over Finite Field of size 7 104 104 sage: P.coordinate_ring() is R 105 105 True … … 144 144 Defn: Structure map 145 145 sage: X.coordinate_ring() 146 Polynomial Ring in x, y, z, w over Rational Field146 Multivariate Polynomial Ring in x, y, z, w over Rational Field 147 147 148 148 Loading and saving: … … 172 172 EXAMPLES: 173 173 sage: ProjectiveSpace(3, GF(19^2,'alpha'), 'abcd').coordinate_ring() 174 Polynomial Ring in a, b, c, d over Finite Field in alpha of size 19^2174 Multivariate Polynomial Ring in a, b, c, d over Finite Field in alpha of size 19^2 175 175 176 176 sage: ProjectiveSpace(3).coordinate_ring() 177 Polynomial Ring in x0, x1, x2, x3 over Integer Ring177 Multivariate Polynomial Ring in x0, x1, x2, x3 over Integer Ring 178 178 179 179 sage: ProjectiveSpace(2, QQ, ['alpha', 'beta', 'gamma']).coordinate_ring() 180 Polynomial Ring in alpha, beta, gamma over Rational Field180 Multivariate Polynomial Ring in alpha, beta, gamma over Rational Field 181 181 """ 182 182 try: … … 240 240 (x*z^2, y^2*z, x*y^2) 241 241 sage: I = X.defining_ideal(); I 242 Ideal (x*z^2, y^2*z, x*y^2) of Polynomial Ring in x, y, z over Rational Field242 Ideal (x*z^2, y^2*z, x*y^2) of Multivariate Polynomial Ring in x, y, z over Rational Field 243 243 sage: I.groebner_basis() 244 244 [x*z^2, y^2*z, x*y^2] -
sage/schemes/generic/spec.py
r4851 r6606 39 39 Spectrum of Univariate Polynomial Ring in x over Rational Field 40 40 sage: Spec(PolynomialRing(QQ, 'x', 3)) 41 Spectrum of Polynomial Ring in x0, x1, x2 over Rational Field41 Spectrum of Multivariate Polynomial Ring in x0, x1, x2 over Rational Field 42 42 sage: X = Spec(PolynomialRing(GF(49,'a'), 3, 'x')); X 43 Spectrum of Polynomial Ring in x0, x1, x2 over Finite Field in a of size 7^243 Spectrum of Multivariate Polynomial Ring in x0, x1, x2 over Finite Field in a of size 7^2 44 44 sage: loads(X.dumps()) == X 45 45 True … … 121 121 Rational Field 122 122 sage: Spec(PolynomialRing(QQ,3, 'x')).coordinate_ring() 123 Polynomial Ring in x0, x1, x2 over Rational Field123 Multivariate Polynomial Ring in x0, x1, x2 over Rational Field 124 124 """ 125 125 return self.__R -
sage/schemes/plane_curves/constructor.py
r4482 r6606 110 110 0 111 111 sage: I = X.defining_ideal(); I 112 Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Polynomial Ring in x, y, z over Rational Field112 Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field 113 113 114 114 EXAMPLE: In three variables, the defining equation must be homogeneous. -
sage/structure/coerce.pyx
r6418 r6606 186 186 t + x + zeta13 187 187 sage: f.parent() 188 Polynomial Ring in t, x over Cyclotomic Field of order 13 and degree 12188 Multivariate Polynomial Ring in t, x over Cyclotomic Field of order 13 and degree 12 189 189 sage: ZZ['x','y'].0 + ~Frac(QQ['y']).0 190 190 (x*y + 1)/y … … 195 195 w + x 196 196 sage: f.parent() 197 Polynomial Ring in w, x, y, z, a over Rational Field197 Multivariate Polynomial Ring in w, x, y, z, a over Rational Field 198 198 sage: ZZ['x,y,z'].0 + ZZ['w,x,z,a'].1 199 199 2*x -
sage/structure/parent.pyx
r6377 r6606 576 576 sage: R.<x,y> = PolynomialRing(QQ, 2) 577 577 sage: R.Hom(QQ) 578 Set of Homomorphisms from Polynomial Ring in x, y over Rational Field to Rational Field578 Set of Homomorphisms from Multivariate Polynomial Ring in x, y over Rational Field to Rational Field 579 579 580 580 Homspaces are defined for very general \sage objects, even elements of familiar rings. -
sage/structure/parent_base.pyx
r5571 r6606 369 369 sage: R.<x,y> = PolynomialRing(QQ, 2) 370 370 sage: R.Hom(QQ) 371 Set of Homomorphisms from Polynomial Ring in x, y over Rational Field to Rational Field371 Set of Homomorphisms from Multivariate Polynomial Ring in x, y over Rational Field to Rational Field 372 372 373 373 Homspaces are defined for very general \sage objects, even elements of familiar rings. -
sage/structure/parent_gens.pyx
r6570 r6606 242 242 sage: R, vars = MPolynomialRing(QQ,3, 'x').objgens() 243 243 sage: R 244 Polynomial Ring in x0, x1, x2 over Rational Field244 Multivariate Polynomial Ring in x0, x1, x2 over Rational Field 245 245 sage: vars 246 246 (x0, x1, x2) … … 309 309 310 310 sage: R = QQ['x,y,abc']; R 311 Polynomial Ring in x, y, abc over Rational Field311 Multivariate Polynomial Ring in x, y, abc over Rational Field 312 312 sage: R.2 313 313 abc
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