Polynomial morphisms for products of projective spaces#
This class builds on the projective space class and its point and morphism classes.
EXAMPLES:
sage: P1xP1.<x,y,u,v> = ProductProjectiveSpaces(QQ, [1, 1])
sage: H = End(P1xP1)
sage: H([x^2*u, y^2*v, x*v^2, y*u^2])
Scheme endomorphism of Product of projective spaces P^1 x P^1 over Rational Field
Defn: Defined by sending (x : y , u : v) to (x^2*u : y^2*v , x*v^2 : y*u^2).
- class sage.schemes.product_projective.morphism.ProductProjectiveSpaces_morphism_ring(parent, polys, check=True)#
Bases:
SchemeMorphism_polynomialThe class of morphisms on products of projective spaces.
The components are projective space morphisms.
EXAMPLES:
sage: T.<x,y,z,w,u> = ProductProjectiveSpaces([2, 1], QQ) sage: H = T.Hom(T) sage: H([x^2, y^2, z^2, w^2, u^2]) Scheme endomorphism of Product of projective spaces P^2 x P^1 over Rational Field Defn: Defined by sending (x : y : z , w : u) to (x^2 : y^2 : z^2 , w^2 : u^2).
- as_dynamical_system()#
Return this endomorphism as a
DynamicalSystem_product_projective.OUTPUT:
EXAMPLES:
sage: Z.<a,b,x,y,z> = ProductProjectiveSpaces([1, 2], ZZ) sage: H = End(Z) sage: f = H([a^3, b^3, x^2, y^2, z^2]) sage: type(f.as_dynamical_system()) # needs sage.schemes <class 'sage.dynamics.arithmetic_dynamics.product_projective_ds.DynamicalSystem_product_projective'>
- global_height(prec=None)#
Return the maximum of the absolute logarithmic heights of the coefficients in any of the coordinate functions of this map.
INPUT:
prec– desired floating point precision (default: default RealField precision).
OUTPUT:
a real number.
Todo
Add functionality for \(\QQbar\), implement function to convert the map defined over \(\QQbar\) to map over a number field.
EXAMPLES:
sage: P1xP1.<x,y,u,v> = ProductProjectiveSpaces([1, 1], ZZ) sage: H = End(P1xP1) sage: f = H([x^2*u, 3*y^2*v, 5*x*v^2, y*u^2]) sage: f.global_height() # needs sage.rings.real_mpfr 1.60943791243410
sage: # needs sage.rings.number_field sage: u = QQ['u'].0 sage: R = NumberField(u^2 - 2, 'v') sage: PP.<x,y,a,b> = ProductProjectiveSpaces([1, 1], R) sage: H = End(PP) sage: O = R.maximal_order() sage: g = H([3*O(u)*x^2, 13*x*y, 7*a*y, 5*b*x + O(u)*a*y]) sage: g.global_height() # needs sage.rings.real_mpfr 2.56494935746154
- is_morphism()#
Return
Trueif this mapping is a morphism of products of projective spaces.For each component space of the codomain of this mapping we consider the subscheme of the domain of this map generated by the corresponding coordinates of the map. This map is a morphism if and only if each of these subschemes has no points.
OUTPUT: Boolean.
EXAMPLES:
sage: Z.<a,b,x,y,z> = ProductProjectiveSpaces([1, 2], ZZ) sage: H = End(Z) sage: f = H([a^2, b^2, x*z - y*z, x^2 - y^2, z^2]) sage: f.is_morphism() # needs sage.libs.singular False
sage: P.<x,y,z,u,v,w> = ProductProjectiveSpaces([2, 2], QQ) sage: H = End(P) sage: f = H([u, v, w, u^2, v^2, w^2]) sage: f.is_morphism() # needs sage.libs.singular True
sage: P.<x,y,z,w,u> = ProductProjectiveSpaces([2, 1], QQ) sage: Q.<a,b,c,d,e> = ProductProjectiveSpaces([1, 2], QQ) sage: H = Hom(P, Q) sage: f = H([x^2, y^2, u^3, w^3, u^3]) sage: f.is_morphism() # needs sage.libs.singular False
- local_height(v, prec=None)#
Return the maximum of the local height of the coefficients in any of the coordinate functions of this map.
INPUT:
v– a prime or prime ideal of the base ring.prec– desired floating point precision (default: default RealField precision).
OUTPUT: A real number.
EXAMPLES:
sage: T.<x,y,z,w,u> = ProductProjectiveSpaces([2, 1], QQ) sage: H = T.Hom(T) sage: f = H([4*x^2 + 3/100*y^2, 8/210*x*y, 1/10000*z^2, 20*w^2, 1/384*u*w]) sage: f.local_height(2) # needs sage.rings.real_mpfr 4.85203026391962
sage: # needs sage.rings.number_field sage: R.<z> = PolynomialRing(QQ) sage: K.<w> = NumberField(z^2 - 5) sage: P.<x,y,a,b> = ProductProjectiveSpaces([1, 1], K) sage: H = Hom(P, P) sage: f = H([2*x^2 + w/3*y^2, 1/w*y^2, a^2, 6*b^2 + 1/9*a*b]) sage: f.local_height(K.ideal(3)) # needs sage.rings.real_mpfr 2.19722457733622