Trac #15378: composition of scheme morphism defined by polynomials

Currently the following fails
{{{
sage: P.<x,y>=ProjectiveSpace(QQ,1)
sage: H=Hom(P,P)
sage: f=H([x^2 -29/16*y^2, y^2])
sage: f*f
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for *:
'SchemeMorphism_polynomial_projective_space_field' and
'SchemeMorphism_polynomial_projective_space_field'
}}}

We just propose a generic implementation.

URL: https://trac.sagemath.org/15378
Reported by: vdelecroix
Ticket author(s): Vincent Delecroix, Ben Hutz
Reviewer(s): Vincent Delecroix

src/sage/schemes/generic/morphism.py

@@ -146,7 +146,7 @@ class SchemeMorphism(Element):
         sub-class of :class:`~sage.structure.element.ModuleElement` and
         :class:`SchemeMorphism`, but Cython would currently confuse cpdef
         attributes of the two base classes. Proper inheritance should be used
-        as soon as this bug is fixed.
+        as soon as this bug is fixed. See :trac:`14711`.
 
     EXAMPLES::
 
@@ -1330,6 +1330,7 @@ def change_ring(self, R, check=True):
                 Defn: Defined on coordinates by sending (x : y) to
                     (x : y)
 
+
         Check that :trac:`16834` is fixed::
 
             sage: A.<x,y,z> = AffineSpace(RR, 3)
@@ -1443,6 +1444,86 @@ def change_ring(self, R, check=True):
                 G.append(f.change_ring(R))
         return(H(G, check))
 
+    def _composition_(self, other, homset):
+        r"""
+        Straightforward implementation of composition for scheme morphisms
+        defined by polynomials.
+
+        TESTS::
+
+            sage: P.<x,y> = ProjectiveSpace(QQ,1)
+            sage: H = Hom(P,P)
+            sage: f = H([x^2 -29/16*y^2, y^2])
+            sage: g = H([y,x+y])
+            sage: h = f*g
+            sage: h
+            Scheme endomorphism of Projective Space of dimension 1 over Rational
+            Field
+              Defn: Defined on coordinates by sending (x : y) to
+                    (-29/16*x^2 - 29/8*x*y - 13/16*y^2 : x^2 + 2*x*y + y^2)
+            sage: p = P((1,3))
+            sage: h(p) == f(g(p))
+            True
+
+            sage: Q = ProjectiveSpace(QQ,2)
+            sage: H2 = Hom(P,Q)
+            sage: h2 = H2([x^2+y^2,x^2,y^2+2*x^2])
+            sage: h2 * f
+            Scheme morphism:
+              From: Projective Space of dimension 1 over Rational Field
+              To:   Projective Space of dimension 2 over Rational Field
+              Defn: Defined on coordinates by sending (x : y) to
+                    (x^4 - 29/8*x^2*y^2 + 1097/256*y^4 : x^4 - 29/8*x^2*y^2 + 841/256*y^4 : 2*x^4 - 29/4*x^2*y^2 + 969/128*y^4)
+
+        ::
+
+            sage: A.<x,y> = AffineSpace(QQ, 2)
+            sage: A1.<z> = AffineSpace(QQ, 1)
+            sage: H = End(A)
+            sage: f = H([x^2+y^2, y^2/x])
+            sage: H1 = Hom(A, A1)
+            sage: g = H1([x + y^2])
+            sage: g*f
+            Scheme morphism:
+              From: Affine Space of dimension 2 over Rational Field
+              To:   Affine Space of dimension 1 over Rational Field
+              Defn: Defined on coordinates by sending (x, y) to
+                    ((x^4 + x^2*y^2 + y^4)/x^2)
+            sage: f*g
+            Traceback (most recent call last):
+            ...
+            TypeError: self (=Scheme endomorphism of Affine Space of dimension 2 over Rational Field
+              Defn: Defined on coordinates by sending (x, y) to
+                    (x^2 + y^2, y^2/x)) domain must equal right (=Scheme morphism:
+              From: Affine Space of dimension 2 over Rational Field
+              To:   Affine Space of dimension 1 over Rational Field
+              Defn: Defined on coordinates by sending (x, y) to
+                    (y^2 + x)) codomain
+
+        Not both defined by polynomials::
+
+            sage: x = polygen(QQ)
+            sage: K.<a> = NumberField(x^2 - 2)
+            sage: p1, p2 = K.Hom(K)
+            sage: R.<x,y> = K[]
+            sage: q1 = R.Hom(R)(p1)
+            sage: A = AffineSpace(R)
+            sage: f1 = A.Hom(A)(q1)
+            sage: g = A.Hom(A)([x^2-y, y+1])
+            sage: g*f1
+            Composite map:
+              From: Affine Space of dimension 2 over Number Field in a with defining polynomial x^2 - 2
+              To:   Affine Space of dimension 2 over Number Field in a with defining polynomial x^2 - 2
+              Defn:   Generic endomorphism of Affine Space of dimension 2 over Number Field in a with defining polynomial x^2 - 2
+                    then
+                      Generic endomorphism of Affine Space of dimension 2 over Number Field in a with defining polynomial x^2 - 2
+        """
+        try:
+            opolys = tuple(other._polys)
+        except AttributeError:
+            return super(SchemeMorphism_polynomial, self)._composition_(other, homset)
+        return homset([p(*opolys) for p in self._polys])
+
 ############################################################################
 # Rational points on schemes, which we view as morphisms determined
 # by coordinates.