Coverage for local/lib/python2.7/site-packages/sage/rings/polynomial/polynomial_ring_homomorphism.pyx : 75%

""" 

Ring homomorphisms from a polynomial ring to another ring 

This module currently implements the canonical ring homomorphism from 

`A[x]` to `B[x]` induced by a ring homomorphism from `A` to `B`. 

.. TODO:: 

Implement homomorphisms from `A[x]` to an arbitrary ring `R`, 

given by a ring homomorphism from `A` to `R` and the image of `x` 

in `R`. 

AUTHORS: 

- Peter Bruin (March 2014): initial version 

""" 

from sage.rings.morphism cimport RingHomomorphism_from_base 

from sage.structure.element cimport Element 

cdef class PolynomialRingHomomorphism_from_base(RingHomomorphism_from_base): 

""" 

The canonical ring homomorphism from `R[x]` to `S[x]` induced by a 

ring homomorphism from `R` to `S`. 

EXAMPLES:: 

sage: QQ['x'].coerce_map_from(ZZ['x']) 

Ring morphism: 

From: Univariate Polynomial Ring in x over Integer Ring 

To: Univariate Polynomial Ring in x over Rational Field 

Defn: Induced from base ring by 

Natural morphism: 

From: Integer Ring 

To: Rational Field 

""" 

cpdef Element _call_(self, x): 

""" 

Evaluate the homomorphism ``self`` at ``x``. 

TESTS:: 

sage: from sage.rings.polynomial.polynomial_ring_homomorphism import PolynomialRingHomomorphism_from_base 

sage: R.<x> = ZZ[] 

sage: S = QQ['x'] 

sage: f = ZZ.hom(QQ) 

sage: F = PolynomialRingHomomorphism_from_base(R.Hom(S), f) 

sage: F(2*x) 

2*x 

sage: A = PolynomialRing(QQ, 'x', sparse=True) 

sage: B = PolynomialRing(RR, 'x', sparse=True) 

sage: g = QQ.hom(RR) 

sage: G = PolynomialRingHomomorphism_from_base(A.Hom(B), g) 

sage: G(A.gen()^1000000) 

1.00000000000000*x^1000000 

""" 

P = self.codomain() 

f = self.underlying_map() 

if P.is_sparse(): 

return P({a: f(b) for a, b in x.dict().iteritems()}) 

else: 

return P([f(b) for b in x]) 

cpdef Element _call_with_args(self, x, args=(), kwds={}): 

""" 

Evaluate ``self`` at ``x`` with additional (keyword) arguments. 

TESTS:: 

sage: from sage.rings.polynomial.polynomial_ring_homomorphism import PolynomialRingHomomorphism_from_base 

sage: R.<x> = ZZ[] 

sage: S = GF(5)['x'] 

sage: f = ZZ.hom(GF(5)) 

sage: F = PolynomialRingHomomorphism_from_base(R.Hom(S), f) 

sage: F(2*x, check=True) 

2*x 

sage: k = GF(49, 'z') 

sage: A = PolynomialRing(GF(7), 'x', sparse=True) 

sage: B = PolynomialRing(k, 'x', sparse=True) 

sage: g = GF(7).hom(k) 

sage: G = PolynomialRingHomomorphism_from_base(A.Hom(B), g) 

sage: G(A.gen()^1000000, True, construct=False) 

x^1000000 

""" 

P = self.codomain() 

f = self.underlying_map() 

if P.is_sparse(): 

return P({a: f(b) for a, b in x.dict().iteritems()}, *args, **kwds) 

else: 

return P([f(b) for b in x], *args, **kwds) 

def is_injective(self): 

r""" 

Return whether this morphism is injective. 

EXAMPLES:: 

sage: R.<x> = ZZ[] 

sage: S.<x> = QQ[] 

sage: R.hom(S).is_injective() 

True 

""" 

return self.underlying_map().is_injective() 

def is_surjective(self): 

r""" 

Return whether this morphism is surjective. 

EXAMPLES:: 

sage: R.<x> = ZZ[] 

sage: S.<x> = Zmod(2)[] 

sage: R.hom(S).is_surjective() 

True 

""" 

return self.underlying_map().is_surjective()