unfinished poly congruence stuff

script/research/ec/hyper/congru.sage

@@ -0,0 +1,33 @@
+R.<x, y, a, b, K> = QQ[]
+a
+
+S1 = R.quotient(x - a)
+
+f = (x - a) + b^2
+assert f(x=a) == b^2
+
+W1 = b
+# This expression is 0 at x = 0
+# => (x - a)^1 is a factor
+assert (W1^2 - f)(x=a) == 0
+# Therefore 0 in the quotient ring
+assert S1(W1^2) == S1(f)
+
+W_prev = W1
+n = 1
+
+S2 = R.quotient((x - a)^(n + 1))
+W_next = W_prev + K*(x - a)^n
+# The term k^2*(x - a)^(2*n) disappears in the quotient ring
+assert S2(K^2*(x - a)^(2*n)) == 0
+assert S2(W_next^2 - f) == S2(W_prev^2 - f + 2*K*(x - a)^n*W_prev)
+
+# Remember from the last step that (x - a)^n is a factor of
+# W_prev^2 - f
+P = (W_prev^2 - f) / (x - a)^n
+k = -P/2
+
+W_next = W_prev + k*(x - a)^n
+assert S2(W_next^2) == S2(W_prev^2 - W_prev^3 + f*W_prev)
+#assert S(W_next^2 - f) == 0
+