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 | 1 | initial version |
Here is a way:
F = GF(23)
A2.<x,y> = AffineSpace(F, 2)
H = Hom(A2, A2)
h0 = H([(x^2 + 1)/x, (x^2*y - y)/x^2])
h1 = H([(x^2 - 2*x + 8)/(x - 2), (x^2*y - 4*x*y - 4*y)/(x^2 - 4*x + 4)])
h2 = H([(x^2 + x + 5)/(x + 1), (x^2*y + 2*x*y - 4*y)/(x^2 + 2*x + 1)])
h3 = H([(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)])
h = h3*h2*h1*h0
Answers to the questions:
$h(2,4)$:
sage: h((2,4))
(10, 12)
The rational map of $h$:
sage: h
Scheme endomorphism of Affine Space of dimension 2 over Finite Field of size 23
Defn: Defined on coordinates by sending (x, y) to
((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))
Its degree:
sage: h.degree()
36
| | 2 | No.2 Revision |
Here is a way:
F = GF(23)
A2.<x,y> = AffineSpace(F, 2)
H = Hom(A2, A2)
h0 = H([(x^2 + 1)/x, (x^2*y - y)/x^2])
h1 = H([(x^2 - 2*x + 8)/(x - 2), (x^2*y - 4*x*y - 4*y)/(x^2 - 4*x + 4)])
h2 = H([(x^2 + x + 5)/(x + 1), (x^2*y + 2*x*y - 4*y)/(x^2 + 2*x + 1)])
h3 = H([(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)])
h = h3*h2*h1*h0
Answers to the questions:
$h(2,4)$:
sage: h((2,4))
(10, 12)
The rational map of $h$:
sage: h
Scheme endomorphism of Affine Space of dimension 2 over Finite Field of size 23
Defn: Defined on coordinates by sending (x, y) to
((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))
Its degree:
sage: h.degree()
36
The defining rational functions:
sage: list(h)
or
sage: tuple(h)
