Deformation Techniques for Efficient Polynomial Equation ... - RISC

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DEFORMATION TECHNIQUES 97 v Since by assumption the polynomial Q approximates Q in QT&t[Y] with precision s, we immediately see that for 1 jr+1 the polynomial M j approximates M j with precision s and that the polynomial 0ls H l approximates the power series 1M 1 = l0 H l with precision s. This implies that the polynomial R (T, Y) approximates R(T, Y) with precision s in QT&t[Y]. Finally, we design a suitable arithmetic network which produces the polynomial R from the input circuit ; representing Q and the coefficient representation of Q(t, Y)=Q (t, Y). The first step in the construction consists in writing down the matrix of the linear mapping . t and to extract 2r independent rows from it. This allows to identify the matrices M t =M (t) and M (T ) as submatrices of the Sylvester matrices of Q(t, Y), Q$(t, Y) and Q (T, Y), Q Y(T, Y). This can be done by means of an arithmetic network of size r O(1) and nonscalar depth O(log r) using the coefficients of Q(t, Y)=Q (t, Y) which represent part of the input. Then we compute the polynomials M j(T ). This can be done by a division-free arithmetic circuit in Q[T] (which extends the input circuit ; ) of size L +(2r+1) O(1) and nonscalar depth * +O(log(2r)). Finally we compute H , 0ls H l and all the coefficients in Y of R using O(r log s) additional arithmetic operations, organized in a circuit of nonscalar depth O(log s). In this way we obtain a division-free arithmetic circuit #~ of required size and nonscalar depth. K Combining our previous results, we are now able to prove Theorem 2. Proof of Theorem 2. Let k :=wlog 2 deg T Px+1. Applying Lemma 5 to the input circuit ;, we obtain a division-free arithmetic circuit # 0 in Q[T, X] of size O(kd 2 n 7 L) and non-scalar depth O(k(log 2 n+*)) which evaluates numerators g 1 := g (k) , ..., g 1 n := g (k) n and a non-zero denominator h :=h (k) for the k-fold iteration of the Newton Hensel operator N k (such that N k (X)=(g F F 1h, ..., g n h) holds). Now, we apply # 0 to compute the matrices g 1 (T, v 1 (M), ..., v n (M)), ..., g n (T, v 1 (M), ..., v n (M)) and h(T, v 1 (M), ..., v n (M)) in Q[T] D_D (compare with (4)). This yields an arithmetic circuit # 1 in Q[T] of size O(kd 2 n 7 D 2 L).
98 HEINTZ ET AL. By means of the well-parallelizable polynomial algorithm of Berkowitz [Ber84] we compute the coefficients of the characteristic polynomial X h # Q[T][Y] of the matrix h(T, v 1 (M), ..., v n (M)). Let with a 0 , ..., a D # Q[T] and let D X h = : l=0 D h*(T, X) := : l=1 a l (T ) Y l a l (T ) h l&1 . Recall from the comments on Lemma 5 that the matrix h(T, v 1 (M), ..., v n (M)) is unimodular in Q[T] D_D . Thus a M 0 is a unit of Q[T] M and in particular we have a 0 (t){0. Hence, the CayleyHamilton Theorem implies that h(T, v 1 (M), ..., v n (M)) &1 = &1 a 0 h*(T, v 1 (M), ..., v n (M)) (17) holds in Q[T] D_D M . We compute now the entries of the matrix h*(T, v 1 (M), ..., v n (M)) by means of an arithmetic circuit # 2 which extends # 1 increasing the size and nonscalar depth of # 1 by D O(1) and O(log D), respectively. We would like to compute the matrix h(T, v 1 (M), ..., v n (M)) &1 by means of formula (17). However, we are unable to do that in Q[T] because this would require division by the polynomial a 0 (T ). Therefore we replace the matrix h(T, v 1 (M), ..., v n (M)) &1 by a suitable approximation in our subsequent argumentation. For this purpose let us use the following identity in QT&t : Thus, the polynomial &1 a 0 (T ) = 1 a 0 (t) l=0\ : a l 0(T )&a 0 (t) . a 0 (t) + a 0 *(T ):= 1 2 a 0 (t) l=0\ k : a l 0(T )&a 0 (t) a 0 (t) + approximates the rational function &1a 0 (T )inQT&t with precision 2 k .
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Page 21 and 22: 90 HEINTZ ET AL. Q approximates Q w
Page 23 and 24: 92 HEINTZ ET AL. Now, recall from (
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Page 37 and 38: 106 HEINTZ ET AL. P 1 , ..., P n yi
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