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: sage.schemes.generic.morphism.SchemeMorphism_polynomial

The 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_producte_projective.

OUTPUT:

  • DynamicalSystem_produce_projective

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())
<class 'sage.dynamics.arithmetic_dynamics.product_projective_ds.DynamicalSystem_product_projective'>
global_height(prec=None)

Returns 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()
1.60943791243410
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()
2.56494935746154
is_morphism()

Returns True if 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()
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()
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()
False
local_height(v, prec=None)

Returns 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)
4.85203026391962
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))
2.19722457733622