On the complexity of the D5 principle

Dahan, Xavier; Schost, Éric; Maza, Marc Moreno; Wu, Wenyuan; Xie, Y.

doi:10.1145/1113439.1113457

Cited by 32 publications

(48 citation statements)

References 4 publications

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“…The best result to date were 4 n δ polylog(δ) operations in F q for modular multiplication [9] and c n δ polylog(δ) for quasi-inverse [5], for some constant c: this is better for fixed n; our result is better when e.g. deg(T i , X i ) = 2 for all i.…”

mentioning

confidence: 76%

Almost linear time operations with triangular sets

Dahan

Maza

Schost

et al. 2011

ACM Commun. Comput. Algebra

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Let F be a perfect field, and let X = X 1 , . . . , X n be indeterminates over F. A (monic) triangular set T = (T 1 , . . . , T n ) is a family of polynomials in F [X] such that for all i, T i is in F[X 1 , . . . , X i ], monic in X i , and reduced modulo T 1 , . . . , T i−1 . The degree of T is the product deg(T 1 , X 1 ) · · · deg(T n , X n ). These objects allow one to solve a variety of problems for systems of polynomial equations, see [7,1,10,6,12]. We are interested here in the complexity of operations modulo a given triangular set T.The first question is modular multiplication: given polynomials A, B reduced modulo T, compute AB mod T.Further operations involve families of triangular sets. The lexicographic Gröbner basis of an ideal I for a given variable order may not be triangular. The workaround is to decompose I as I = I 1 ∩· · ·∩I s , with pairwise coprime I j , where each I j admits a triangular basis. The decomposition is in general not unique, but there exists a canonical choice, the equiprojectable decomposition [4].That said, the most useful notion of "inversion" is quasi-inverses: given A reduced modulo T, we decompose the ideal T as I 0 ∩ I 1 , where A is zero modulo I 0 and invertible modulo I 1 ; the output is the equiprojectable decompositions of I 0 , I 1 , and the inverse of A modulo the triangular sets that define I 1 . The next question is change of order: starting from T, we output the equiprojectable decomposition of the ideal T , for a new order on the variables. The last question starts from a family T(1) , . . . , T (r) which generate pairwise coprime ideals; our output is the equiprojectable decomposition of the ideal they generate.The following theorem provides quasi-linear time results for these questions. These results are valid over a finite field, with costs given in a boolean RAM model; the algorithms are Las Vegas. The main idea is to introduce a primitive element and change representation, as most problems above can be solved easily in univariate situations. The change of representation is done using algorithms for modular composition [3] and power projection [13], but in multivariate setting. In [8], Kedlaya and Umans introduced quasi-linear time algorithms for the univariate versions of these problems; our core technical ingredients are multivariate versions of their algorithms.Theorem 1. For any ε > 0, there exists a constant c ε such that the following problems can be solved using an expected c ε δ 1+ε log(q) log log(q) 5 bit operations:1

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confidence: 76%

Almost linear time operations with triangular sets

Dahan

Maza

Schost

et al. 2011

ACM Commun. Comput. Algebra

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“…is equivalent to n m . In this case, there exist algorithms of cost O˜(δm) = O˜(n m ) for multiplication and inversion (when possible) in Am [17,30]. Here, and everywhere else in this paper, the O˜notation indicates the omission of logarithmic factors.…”

Section: Lp (T ) :=mentioning

confidence: 99%

Algorithms for the universal decomposition algebra

Lebreton

Schost

2012

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

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Let k be a field and let f ∈ k [T ] be a polynomial of degree n. The universal decomposition algebra A is the quotient of k [X1, . . . , Xn] by the ideal of symmetric relations (those polynomials that vanish on all permutations of the roots of f ). We show how to obtain efficient algorithms to compute in A. We use a univariate representation of A, i.e. an isomorphism of the form A k[T ]/Q(T ), since in this representation, arithmetic operations in A are known to be quasi-optimal. We give details for two related algorithms, to find the isomorphism above, and to compute the characteristic polynomial of any element of A.

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“…We rely on classical, thus quadratic, algorithms for computing GCDs modulo zero-dimensional regular chains [9]. Our motivation is practical: we aim at handling problem sizes to which the asymptotically fast GCDs of [4] are not likely to apply. Even if they would, we do not have yet implementations for these fast GCDs.…”

Section: A Complexity Analysismentioning

confidence: 99%

“…Then we use the augment refinement method of [1] to compute a polynomial GCD-free basis over a DPF. Following the inductive process applied in [4], we achieve the complexity result of Theorem 1. Recall that an arithmetic time T → A n (deg 1 T, .…”

Section: A Complexity Analysismentioning

confidence: 99%