Lecture Notes on Computer Algebra
نویسنده
چکیده
These notes record seven lectures given in the computer algebra course in the fall of 2004. The theory of subresultants is not required for the final exam due to its complicated constructions. 1 Chinese remainder algorithm 1.1 Modular arithmetic Let m be a positive integer greater than one. We discuss basic operations in Z/(m). The elements in Z/(m) can be represented in two ways: • {0, 1, . . . , m− 1} (nonnegative representation); • { a ∈ Z | −m2 < a ≤ m 2 } (symmetric representation). Let us fix a representation Zm e.g. the nonnegative representation. Via division we have a canonical simplifier Hm from Z onto Zm. Proposition 1 For n ∈ Z with L(n) ≥ L(m), the time for computing Hm(n) is dominated by O(L(m)(L(n)−L(m)+1)), where L(n) denotes the length of the integer n. Proof. It follows from the cost estimation of integral division (see Theorem 2.1.7 in [8]). For two elements a, b in Zm, we compute a + b, a − b, and ab as if they were integers, and then apply Hm to the results. Thus, the costs for computing (a+b) and (a−b) are dominated by O(L(m)), and the cost for ab dominated by O(L(m)2). The latter estimation kills any hope to use fast multiplications for modular multiplication. A particular operation in Zm is the inversion. Given a ∈ Zm with gcd(a,m) = 1, compute an element b ∈ Zm such that ab ≡ 1 mod m.
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