# 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)

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 flattening 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.SpecializationMorphism(domain, D)

Morphisms to specialize parameters in (stacked) polynomial rings

EXAMPLES:

sage: R.<c> = PolynomialRing(QQ)
sage: S.<x,y,z> = PolynomialRing(R)
sage: D = dict({c:1})
sage: from sage.rings.polynomial.flatten import SpecializationMorphism
sage: f = SpecializationMorphism(S, D)
sage: g = f(x^2 + c*y^2 - z^2); g
x^2 + y^2 - z^2
sage: g.parent()
Multivariate Polynomial Ring in x, y, z over Rational Field

sage: R.<c> = PolynomialRing(QQ)
sage: S.<z> = PolynomialRing(R)
sage: from sage.rings.polynomial.flatten import SpecializationMorphism
sage: xi = SpecializationMorphism(S, {c:0}); xi
Specialization morphism:
From: Univariate Polynomial Ring in z over Univariate Polynomial Ring in c over Rational Field
To:   Univariate Polynomial Ring in z over Rational Field
sage: xi(z^2+c)
z^2

sage: R1.<u,v> = PolynomialRing(QQ)
sage: R2.<a,b,c> = PolynomialRing(R1)
sage: S.<x,y,z> = PolynomialRing(R2)
sage: D = dict({a:1, b:2, x:0, u:1})
sage: from sage.rings.polynomial.flatten import SpecializationMorphism
sage: xi = SpecializationMorphism(S, D); xi
Specialization morphism:
From: Multivariate Polynomial Ring in x, y, z over Multivariate Polynomial Ring in a, b, c over Multivariate Polynomial Ring in u, v over Rational Field
To:   Multivariate Polynomial Ring in y, z over Univariate Polynomial Ring in c over Univariate Polynomial Ring in v over Rational Field
sage: xi(a*(x*z+y^2)*u+b*v*u*(x*z+y^2)*y^2*c+c*y^2*z^2)
2*v*c*y^4 + c*y^2*z^2 + y^2

class sage.rings.polynomial.flatten.UnflatteningMorphism(domain, codomain)

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