My actual aim is to evaluate the composition of $n$ maps $h(x, y)=f_n(x, y)\circ \cdots \circ f_1(x, y)$ over the finite field $GF(q)$ for a prime $q$ at specified point $(X, Y)$. The map $h(x, y)$ is specified as a composite map. The maps $f_i(x, y)$s are not accessible, so they can not be evaluated at $(X, Y)$, recursively. Here is a toy example.
from sage.categories.morphism import SetMorphism
F = GF(23)
R.<x, y> = PolynomialRing(F)
# This part is not accessible
f0 = ((x^2 + 1)/x, (x^2*y - y)/x^2)
f1 = ((x^2 - 2*x + 8)/(x - 2), (x^2*y - 4*x*y - 4*y)/(x^2 - 4*x + 4))
f2 = ((x^2 + x + 5)/(x + 1), (x^2*y + 2*x*y - 4*y)/(x^2 + 2*x + 1))
f3 = ((x^3 + 11*x^2 - x - 4)/(x^2 + 11*x - 10), (x^3*y + 5*x^2*y + 7*x*y - 9*y)/(x^3 + 5*x^2 - 7*x - 9))
h0 = SetMorphism(Hom(F, F), f0)
h1 = SetMorphism(Hom(F, F), f1)
h2 = SetMorphism(Hom(F, F), f2)
h3 = SetMorphism(Hom(F, F), f3)
# This part is not accessible
h = h0*h1*h2*h3
print (h)
print (h(2, 4))
This outputs
Composite map:
From: Finite Field of size 23
To: Finite Field of size 23
Defn: Generic endomorphism of Finite Field of size 23
then
Generic endomorphism of Finite Field of size 23
then
Generic endomorphism of Finite Field of size 23
then
Generic endomorphism of Finite Field of size 23
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
Cell In[1], line 23
21 h = h0*h1*h2*h3
22 print (h)
---> 23 print (h(Integer(2), Integer(4)))
File /home/sc_serv/sage/src/sage/categories/map.pyx:825, in sage.categories.map.Map.__call__()
823 if not args and not kwds:
824 return self._call_(x)
--> 825 return self._call_with_args(x, args, kwds)
826
827 cpdef Element _call_(self, x):
File /home/sc_serv/sage/src/sage/categories/map.pyx:1765, in
sage.categories.map.FormalCompositeMap._call_with_args()
1763 """
1764 for f in self.__list[:-1]:
-> 1765 x = f._call_(x)
1766 return self.__list[-1]._call_with_args(x, args, kwds)
1767
File /home/sc_serv/sage/src/sage/categories/morphism.pyx:593, in sage.categories.morphism.SetMorphism._call_()
591 self._function = function
592
--> 593 cpdef Element _call_(self, x):
594 """
595 INPUT:
File /home/sc_serv/sage/src/sage/categories/morphism.pyx:612, in sage.categories.morphism.SetMorphism._call_()
610 3
611 """
--> 612 return self._function(x)
613
614 cpdef Element _call_with_args(self, x, args=(), kwds={}):
TypeError: 'tuple' object is not callable
But, the composed rational map should look like,
h = ((x^24 + 2*x^23 + 2*x^22 + 4*x^21 + 6*x^20 + 11*x^19 - 5*x^18 - 2*x^17 + 9*x^16 + 6*x^15 - 4*x^14 + 11*x^13 + 11*x^11 - 4*x^10 + 6*x^9 + 9*x^8 - 2*x^7 - 5*x^6 + 11*x^5 + 6*x^4 + 4*x^3 + 2*x^2 + 2*x + 1)/(x^23 + 2*x^22 + 2*x^21 + 4*x^20 + 6*x^19 + 11*x^18 - 5*x^17 - 2*x^16 + 9*x^15 + 6*x^14 - 4*x^13 + 9*x^12 - 4*x^11 + 6*x^10 + 9*x^9 - 2*x^8 - 5*x^7 + 11*x^6 + 6*x^5 + 4*x^4 + 2*x^3 + 2*x^2 + x), (x^35*y + 3*x^34*y - 7*x^33*y - 3*x^32*y + x^31*y + 8*x^30*y + x^29*y + 3*x^28*y + 3*x^27*y + 2*x^26*y + 3*x^25*y + 7*x^24*y - x^23*y - 2*x^21*y + 9*x^20*y + 11*x^19*y + x^18*y + x^17*y + 11*x^16*y + 9*x^15*y - 2*x^14*y - x^12*y + 7*x^11*y + 3*x^10*y + 2*x^9*y + 3*x^8*y + 3*x^7*y + x^6*y + 8*x^5*y + x^4*y - 3*x^3*y - 7*x^2*y + 3*x*y + y)/(x^35 + 3*x^34 - 7*x^33 - 3*x^32 + x^31 + 8*x^30 + x^29 + 3*x^28 + 3*x^27 + 2*x^26 + 3*x^25 + 4*x^24 - 10*x^23 + 10*x^22 - 3*x^21 - x^20 - 2*x^19 + 2*x^18 + x^17 + 3*x^16 - 10*x^15 + 10*x^14 - 4*x^13 - 3*x^12 - 2*x^11 - 3*x^10 - 3*x^9 - x^8 - 8*x^7 - x^6 + 3*x^5 + 7*x^4 - 3*x^3 - x^2))
My questions are,
- How to evaluate the map $h$ on the point $(X=2, Y= 4)$. i.e., how to find $h(2, 4)$?
- How to find the rational map of $h$ and its degree?
I would be thankful if anyone suggests the method.
