A Characteristic Property of the Algebra C(Q)B by Karakhanyan M.I., Khor'kova T.A.

By Karakhanyan M.I., Khor'kova T.A.

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That C = BA is fully and uniquely determined by M C C This terminates the correctness proof. The claimed complexity derives immediately from Propositions 2 and 3 that are proved in the next subsections. 1 k! (k − i)! P ˜ m,n Proposition 2 Algorithm Eval computes M in M(mn)+ O(mn) ops. Proof. The series exp(X) mod X n+1 and the factorials 1, . . , n! are computed by recurrence relations in O(n) ops. The computation of S can be done in M(s ) for the size s P ˜ m,n of the corresponding diagonal of M .

Note that Cu,v (X, Y ) has bidegree at most (2 n/p , 2n). To perform Step 3, each Cu,v (X p , θ)X v is first rewrit˜u,v (X p , θ) by computing 2 n/p + 1 shifts of ten as X v C polynomials of degree at most 2n. This can be done in ` ´ O pn M(n) log n ops. Finally, O(pn2 ) ops. are sufficient to Pp−1 u Pp−1 v ˜ p put C = u=0 X v=0 X Cu,v (X , θ) in canonical form. Summarizing, we have just proved: min(d−i,r) X (T )i X +i i ∂ . =− min(i,r) Figure 6: Interpolation in K[X] ∂ . Proposition 3 Interpol computes P in M(dr) + O(dr) ops.

Input: A, B ∈ K[X] θ , with char(K) = p > 0. Output: their product C = BA. Pp−1 1. Rewrite A and B as A = v=0 Av (X p , θ)X v Pp−1 u p and B = u=0 X Bu (X , θ). 2. Compute the commutative bivariate products Cu,v = Bu Av , for 0 ≤ u, v < p. P u p v 3. Write p−1 u,v=0 X Cu,v (X , θ)X in canonical form; return it. Interpol(M ) Input: M ∈ K(d+1)×(r+1) . P ˜ d,r Output: P ∈ Wd,r such that M = M. 1. Divide the kth column of M by k!. 2. For each −r ≤ ≤ d, compute the product T = « „ min(d− ,r) P Mk+ ,k X k exp(−X) mod X min(d− ,r)+1 .

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