Class to flatten polynomial rings over polynomial ring¶
For example QQ['a','b'],['x','y']
flattens to QQ['a','b','x','y']
.
EXAMPLES:
sage: R = QQ['x']['y']['s','t']['X']
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: phi = FlatteningMorphism(R); phi
Flattening morphism:
From: Univariate Polynomial Ring in X over Multivariate Polynomial Ring in s, t over Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
To: Multivariate Polynomial Ring in x, y, s, t, X over Rational Field
sage: phi('x*y*s + t*X').parent()
Multivariate Polynomial Ring in x, y, s, t, X over Rational Field
Authors:
Vincent Delecroix, Ben Hutz (July 2016): initial implementation
-
class
sage.rings.polynomial.flatten.
FlatteningMorphism
(domain)¶ Bases:
sage.categories.morphism.Morphism
EXAMPLES:
sage: R = QQ['a','b']['x','y','z']['t1','t2'] sage: from sage.rings.polynomial.flatten import FlatteningMorphism sage: f = FlatteningMorphism(R) sage: f.codomain() Multivariate Polynomial Ring in a, b, x, y, z, t1, t2 over Rational Field sage: p = R('(a+b)*x + (a^2-b)*t2*(z+y)') sage: p ((a^2 - b)*y + (a^2 - b)*z)*t2 + (a + b)*x sage: f(p) a^2*y*t2 + a^2*z*t2 - b*y*t2 - b*z*t2 + a*x + b*x sage: f(p).parent() Multivariate Polynomial Ring in a, b, x, y, z, t1, t2 over Rational Field
Also works when univariate polynomial ring are involved:
sage: R = QQ['x']['y']['s','t']['X'] sage: from sage.rings.polynomial.flatten import FlatteningMorphism sage: f = FlatteningMorphism(R) sage: f.codomain() Multivariate Polynomial Ring in x, y, s, t, X over Rational Field sage: p = R('((x^2 + 1) + (x+2)*y + x*y^3)*(s+t) + x*y*X') sage: p x*y*X + (x*y^3 + (x + 2)*y + x^2 + 1)*s + (x*y^3 + (x + 2)*y + x^2 + 1)*t sage: f(p) x*y^3*s + x*y^3*t + x^2*s + x*y*s + x^2*t + x*y*t + x*y*X + 2*y*s + 2*y*t + s + t sage: f(p).parent() Multivariate Polynomial Ring in x, y, s, t, X over Rational Field
-
section
()¶ Inverse of this flattenning morphism.
EXAMPLES:
sage: R = QQ['a','b','c']['x','y','z'] sage: from sage.rings.polynomial.flatten import FlatteningMorphism sage: h = FlatteningMorphism(R) sage: h.section() Unflattening morphism: From: Multivariate Polynomial Ring in a, b, c, x, y, z over Rational Field To: Multivariate Polynomial Ring in x, y, z over Multivariate Polynomial Ring in a, b, c over Rational Field
sage: R = ZZ['a']['b']['c'] sage: from sage.rings.polynomial.flatten import FlatteningMorphism sage: FlatteningMorphism(R).section() Unflattening morphism: From: Multivariate Polynomial Ring in a, b, c over Integer Ring To: Univariate Polynomial Ring in c over Univariate Polynomial Ring in b over Univariate Polynomial Ring in a over Integer Ring
-
-
class
sage.rings.polynomial.flatten.
UnflatteningMorphism
(domain, codomain)¶ Bases:
sage.categories.morphism.Morphism
Inverses for
FlatteningMorphism
EXAMPLES:
sage: R = QQ['c','x','y','z'] sage: S = QQ['c']['x','y','z'] sage: from sage.rings.polynomial.flatten import UnflatteningMorphism sage: f = UnflatteningMorphism(R, S) sage: g = f(R('x^2 + c*y^2 - z^2'));g x^2 + c*y^2 - z^2 sage: g.parent() Multivariate Polynomial Ring in x, y, z over Univariate Polynomial Ring in c over Rational Field
sage: R = QQ['a','b', 'x','y'] sage: S = QQ['a','b']['x','y'] sage: from sage.rings.polynomial.flatten import UnflatteningMorphism sage: UnflatteningMorphism(R, S) Unflattening morphism: From: Multivariate Polynomial Ring in a, b, x, y over Rational Field To: Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a, b over Rational Field