Cross-post from: https://ask.sagemath.org/question/80862/implementing-linear-maps-and-their-inverses-in-sagemath-and-magma/
I am currently trying to implement the linear maps in this post: https://math.stackexchange.com/questions/5015461/computing-the-inverses-of-two-linear-maps in Magma and SageMath. So far I have successfully implemented the maps in Magma and am currently trying to implement `X` in SageMath. My Magma code for `X`: ``` k := 8; Fq := GF(2^k); F := FunctionField(Fq); places := Places(F, 1); G := &+[places[i]: i in [2..k]]; B1 := Basis(G); places := [places[i]: i in [k+1..2*k]]; X_matrix := Matrix([[Evaluate(f, P): P in places]: f in B1]); X := function(f) return Vector([Evaluate(f, P) : P in places]); end function; X_inv := function(v) coeffs:= Solution(X_matrix, v); return &+[coeffs[i] * B1[i]: i in [1..k]]; end function; ``` My SageMath code for `X`: ``` k = 8 m = 16 Fq = GF(2^k) F.<x> = FunctionField(Fq) places = F.places() G = sum(places[i] for i in range(1,k)) B1 = list(reversed(G.basis_function_space())) places = places[k:2*k] X_matrix = matrix([[f.evaluate(p) for p in places] for f in B1]) def X(f): return vector([f.evaluate(p) for p in places]) def X_inv(v): coeffs = X_matrix.solve_right(v) return sum(coeff * b for coeff, b in zip(coeffs, B1)) ``` `X` does not work correctly in SageMath. I have noticed that `v`, `B1`, and `X_matrix` are symbolically represented in Magma with `F`'s generator inside `X_inv` rather than `Fq`'s as they are outside. `v`: outside `X_inv`: ``` (1 Fq.1^25 Fq.1^50 Fq.1^198 Fq.1^223 Fq.1^238 Fq.1^104 Fq.1^75) ``` inside `X_inv`: ``` (1 $.1^25 $.1^50 $.1^198 $.1^223 $.1^238 $.1^104 $.1^75) ``` `B1`: outside `X_inv`: ``` [ ($.1)^-1 * ($.1 + Fq.1)^-1 * ($.1 + Fq.1^2)^-1 * ($.1 + Fq.1^3)^-1 * ($.1 + Fq.1^4)^-1 * ($.1 + Fq.1^5)^-1 * ($.1 + Fq.1^6)^-1, ($.1 + Fq.1)^-1 * ($.1 + Fq.1^2)^-1 * ($.1 + Fq.1^3)^-1 * ($.1 + Fq.1^4)^-1 * ($.1 + Fq.1^5)^-1 * ($.1 + Fq.1^6)^-1, ($.1) * ($.1 + Fq.1)^-1 * ($.1 + Fq.1^2)^-1 * ($.1 + Fq.1^3)^-1 * ($.1 + Fq.1^4)^-1 * ($.1 + Fq.1^5)^-1 * ($.1 + Fq.1^6)^-1, ($.1)^2 * ($.1 + Fq.1)^-1 * ($.1 + Fq.1^2)^-1 * ($.1 + Fq.1^3)^-1 * ($.1 + Fq.1^4)^-1 * ($.1 + Fq.1^5)^-1 * ($.1 + Fq.1^6)^-1, ($.1)^3 * ($.1 + Fq.1)^-1 * ($.1 + Fq.1^2)^-1 * ($.1 + Fq.1^3)^-1 * ($.1 + Fq.1^4)^-1 * ($.1 + Fq.1^5)^-1 * ($.1 + Fq.1^6)^-1, ($.1)^4 * ($.1 + Fq.1)^-1 * ($.1 + Fq.1^2)^-1 * ($.1 + Fq.1^3)^-1 * ($.1 + Fq.1^4)^-1 * ($.1 + Fq.1^5)^-1 * ($.1 + Fq.1^6)^-1, ($.1)^5 * ($.1 + Fq.1)^-1 * ($.1 + Fq.1^2)^-1 * ($.1 + Fq.1^3)^-1 * ($.1 + Fq.1^4)^-1 * ($.1 + Fq.1^5)^-1 * ($.1 + Fq.1^6)^-1, ($.1)^6 * ($.1 + Fq.1)^-1 * ($.1 + Fq.1^2)^-1 * ($.1 + Fq.1^3)^-1 * ($.1 + Fq.1^4)^-1 * ($.1 + Fq.1^5)^-1 * ($.1 + Fq.1^6)^-1 ] ``` inside `X_inv`: ``` [ ($.1)^-1 * ($.1 + $.1)^-1 * ($.1 + $.1^2)^-1 * ($.1 + $.1^3)^-1 * ($.1 + $.1^4)^-1 * ($.1 + $.1^5)^-1 * ($.1 + $.1^6)^-1, ($.1 + $.1)^-1 * ($.1 + $.1^2)^-1 * ($.1 + $.1^3)^-1 * ($.1 + $.1^4)^-1 * ($.1 + $.1^5)^-1 * ($.1 + $.1^6)^-1, ($.1) * ($.1 + $.1)^-1 * ($.1 + $.1^2)^-1 * ($.1 + $.1^3)^-1 * ($.1 + $.1^4)^-1 * ($.1 + $.1^5)^-1 * ($.1 + $.1^6)^-1, ($.1)^2 * ($.1 + $.1)^-1 * ($.1 + $.1^2)^-1 * ($.1 + $.1^3)^-1 * ($.1 + $.1^4)^-1 * ($.1 + $.1^5)^-1 * ($.1 + $.1^6)^-1, ($.1)^3 * ($.1 + $.1)^-1 * ($.1 + $.1^2)^-1 * ($.1 + $.1^3)^-1 * ($.1 + $.1^4)^-1 * ($.1 + $.1^5)^-1 * ($.1 + $.1^6)^-1, ($.1)^4 * ($.1 + $.1)^-1 * ($.1 + $.1^2)^-1 * ($.1 + $.1^3)^-1 * ($.1 + $.1^4)^-1 * ($.1 + $.1^5)^-1 * ($.1 + $.1^6)^-1, ($.1)^5 * ($.1 + $.1)^-1 * ($.1 + $.1^2)^-1 * ($.1 + $.1^3)^-1 * ($.1 + $.1^4)^-1 * ($.1 + $.1^5)^-1 * ($.1 + $.1^6)^-1, ($.1)^6 * ($.1 + $.1)^-1 * ($.1 + $.1^2)^-1 * ($.1 + $.1^3)^-1 * ($.1 + $.1^4)^-1 * ($.1 + $.1^5)^-1 * ($.1 + $.1^6)^-1 ] ``` `X_matrix`: outside `X_inv`: ``` [ Fq.1^10 Fq.1^177 Fq.1^26 Fq.1^128 Fq.1^206 Fq.1^98 Fq.1^161 Fq.1^173] [ Fq.1^17 Fq.1^185 Fq.1^35 Fq.1^138 Fq.1^217 Fq.1^110 Fq.1^174 Fq.1^187] [ Fq.1^24 Fq.1^193 Fq.1^44 Fq.1^148 Fq.1^228 Fq.1^122 Fq.1^187 Fq.1^201] [ Fq.1^31 Fq.1^201 Fq.1^53 Fq.1^158 Fq.1^239 Fq.1^134 Fq.1^200 Fq.1^215] [ Fq.1^38 Fq.1^209 Fq.1^62 Fq.1^168 Fq.1^250 Fq.1^146 Fq.1^213 Fq.1^229] [ Fq.1^45 Fq.1^217 Fq.1^71 Fq.1^178 Fq.1^6 Fq.1^158 Fq.1^226 Fq.1^243] [ Fq.1^52 Fq.1^225 Fq.1^80 Fq.1^188 Fq.1^17 Fq.1^170 Fq.1^239 Fq.1^2] [ Fq.1^59 Fq.1^233 Fq.1^89 Fq.1^198 Fq.1^28 Fq.1^182 Fq.1^252 Fq.1^16] ``` inside `X_inv`: ``` [ $.1^10 $.1^177 $.1^26 $.1^128 $.1^206 $.1^98 $.1^161 $.1^173] [ $.1^17 $.1^185 $.1^35 $.1^138 $.1^217 $.1^110 $.1^174 $.1^187] [ $.1^24 $.1^193 $.1^44 $.1^148 $.1^228 $.1^122 $.1^187 $.1^201] [ $.1^31 $.1^201 $.1^53 $.1^158 $.1^239 $.1^134 $.1^200 $.1^215] [ $.1^38 $.1^209 $.1^62 $.1^168 $.1^250 $.1^146 $.1^213 $.1^229] [ $.1^45 $.1^217 $.1^71 $.1^178 $.1^6 $.1^158 $.1^226 $.1^243] [ $.1^52 $.1^225 $.1^80 $.1^188 $.1^17 $.1^170 $.1^239 $.1^2] [ $.1^59 $.1^233 $.1^89 $.1^198 $.1^28 $.1^182 $.1^252 $.1^16] ``` I have figured out how to symbolically represent `X_matrix` and `v` similarly in SageMath but am unsure how to do so with `B1`. `B1` in SageMath: ``` [ 1/(x^7 + (z8^6 + z8^5 + z8^4 + z8^3 + z8^2 + z8)*x^6 + z8^2*x^5 + (z8^7 + z8^4 + z8^3 + z8^2 + z8)*x^4 + (z8^5 + z8^4 + z8^3 + z8)*x^3 + (z8^5 + z8^4 + 1)*x^2 + (z8^6 + z8^5 + z8^4 + z8^2 + 1)*x), 1/(x^6 + (z8^6 + z8^5 + z8^4 + z8^3 + z8^2 + z8)*x^5 + z8^2*x^4 + (z8^7 + z8^4 + z8^3 + z8^2 + z8)*x^3 + (z8^5 + z8^4 + z8^3 + z8)*x^2 + (z8^5 + z8^4 + 1)*x + z8^6 + z8^5 + z8^4 + z8^2 + 1), x/(x^6 + (z8^6 + z8^5 + z8^4 + z8^3 + z8^2 + z8)*x^5 + z8^2*x^4 + (z8^7 + z8^4 + z8^3 + z8^2 + z8)*x^3 + (z8^5 + z8^4 + z8^3 + z8)*x^2 + (z8^5 + z8^4 + 1)*x + z8^6 + z8^5 + z8^4 + z8^2 + 1), x^2/(x^6 + (z8^6 + z8^5 + z8^4 + z8^3 + z8^2 + z8)*x^5 + z8^2*x^4 + (z8^7 + z8^4 + z8^3 + z8^2 + z8)*x^3 + (z8^5 + z8^4 + z8^3 + z8)*x^2 + (z8^5 + z8^4 + 1)*x + z8^6 + z8^5 + z8^4 + z8^2 + 1), x^3/(x^6 + (z8^6 + z8^5 + z8^4 + z8^3 + z8^2 + z8)*x^5 + z8^2*x^4 + (z8^7 + z8^4 + z8^3 + z8^2 + z8)*x^3 + (z8^5 + z8^4 + z8^3 + z8)*x^2 + (z8^5 + z8^4 + 1)*x + z8^6 + z8^5 + z8^4 + z8^2 + 1), x^4/(x^6 + (z8^6 + z8^5 + z8^4 + z8^3 + z8^2 + z8)*x^5 + z8^2*x^4 + (z8^7 + z8^4 + z8^3 + z8^2 + z8)*x^3 + (z8^5 + z8^4 + z8^3 + z8)*x^2 + (z8^5 + z8^4 + 1)*x + z8^6 + z8^5 + z8^4 + z8^2 + 1), x^5/(x^6 + (z8^6 + z8^5 + z8^4 + z8^3 + z8^2 + z8)*x^5 + z8^2*x^4 + (z8^7 + z8^4 + z8^3 + z8^2 + z8)*x^3 + (z8^5 + z8^4 + z8^3 + z8)*x^2 + (z8^5 + z8^4 + 1)*x + z8^6 + z8^5 + z8^4 + z8^2 + 1), x^6/(x^6 + (z8^6 + z8^5 + z8^4 + z8^3 + z8^2 + z8)*x^5 + z8^2*x^4 + (z8^7 + z8^4 + z8^3 + z8^2 + z8)*x^3 + (z8^5 + z8^4 + z8^3 + z8)*x^2 + (z8^5 + z8^4 + 1)*x + z8^6 + z8^5 + z8^4 + z8^2 + 1) ] ``` Any help would be much appreciated. -- You received this message because you are subscribed to the Google Groups "sage-support" group. 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