82 lines
2.5 KiB
Python
82 lines
2.5 KiB
Python
def witt_pol(X, p, n):
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n = len(X)
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return sum(p^i*X[i]^(p^(n-i-1)) for i in range(0, n))
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def witt_sum(p, n):
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variables = ''
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for i in range(0, n+1):
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variables += 'X' + str(i) + ','
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for i in range(0, n+1):
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variables += 'Y' + str(i)
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if i!=n:
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variables += ','
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RQ = PolynomialRing(QQ, variables, 2*(n+1))
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X = RQ.gens()[:n+1]
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Y = RQ.gens()[n+1:]
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Rpx.<x> = PolynomialRing(GF(p), 1)
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RQx.<x> = PolynomialRing(QQ, 1)
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if n == 0:
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return X[0] + Y[0]
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WS = []
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for k in range(0, n):
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aux = witt_sum(p, k)
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Rold = aux.parent()
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Xold = Rold.gens()[:k+1]
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Yold = Rold.gens()[k+1:]
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WS+= [aux.subs({Xold[i] : X[i] for i in range(0, k)} | {Yold[i] : Y[i] for i in range(0, k)})]
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return 1/p^n*(witt_pol(X[:n+1], p, n) + witt_pol(Y[:n+1], p, n) - sum(p^k*WS[k]^(p^(n-k)) for k in range(0, n)))
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def witt_prod(p, n):
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variables = ''
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for i in range(0, n+1):
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variables += 'X' + str(i) + ','
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for i in range(0, n+1):
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variables += 'Y' + str(i)
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if i!=n:
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variables += ','
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RQ = PolynomialRing(QQ, variables, 2*(n+1))
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X = RQ.gens()[:n+1]
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Y = RQ.gens()[n+1:]
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Rpx.<x> = PolynomialRing(GF(p), 1)
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RQx.<x> = PolynomialRing(QQ, 1)
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if n == 0:
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return X[0] * Y[0]
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WP = []
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for k in range(0, n):
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aux = witt_prod(p, k)
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Rold = aux.parent()
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Xold = Rold.gens()[:k+1]
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Yold = Rold.gens()[k+1:]
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WP+= [aux.subs({Xold[i] : X[i] for i in range(0, k)} | {Yold[i] : Y[i] for i in range(0, k)})]
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return 1/p^n*(witt_pol(X[:n+1], p, n) * witt_pol(Y[:n+1], p, n) - sum(p^k*WP[k]^(p^(n-k)) for k in range(0, n)))
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p = 5
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n = 1
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variables = ''
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for i in range(0, n+1):
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variables += 'X' + str(i) + ','
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for i in range(0, n+1):
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variables += 'Y' + str(i)
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if i!=n:
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variables += ','
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RQ = PolynomialRing(QQ, variables, 2*(n+1))
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X = RQ.gens()[:n+1]
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Y = RQ.gens()[n+1:]
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WS = []
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WP = []
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for k in range(0, n+1):
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aux1 = witt_sum(p, k)
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aux2 = witt_prod(p, k)
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Rold = aux1.parent()
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aux2 = Rold(aux2)
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Xold = Rold.gens()[:k+1]
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Yold = Rold.gens()[k+1:]
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WS+= [RQ(aux1.subs({Xold[i] : X[i] for i in range(0, k)} | {Yold[i] : Y[i] for i in range(0, k)}))]
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WP+= [RQ(aux2.subs({Xold[i] : X[i] for i in range(0, k)} | {Yold[i] : Y[i] for i in range(0, k)}))]
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Rp2n = PolynomialRing(GF(p), variables, 2*(n+1))
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X = Rp2n.gens()[:n+1]
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Y = Rp2n.gens()[n+1:]
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WS = [Rp2n(a) for a in WS]
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WP = [Rp2n(a) for a in WP] |