Sparse FGLM algorithms
Given a zero-dimensional ideal I⊂K[x1,…,xn] of degree D, the transformation of the ordering of its Gröbner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms...
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| Veröffentlicht in: | Journal of symbolic computation Jg. 80; H. 3; S. 538 - 569 |
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01.05.2017
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| Abstract | Given a zero-dimensional ideal I⊂K[x1,…,xn] of degree D, the transformation of the ordering of its Gröbner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering.
The main contributions of this paper are several efficient methods for the change of ordering which take advantage of the sparsity of multiplication matrices in the classical FGLM algorithm. Combining all these methods, we propose a deterministic top-level algorithm that automatically detects which method to use depending on the input. As a by-product, we have a fast implementation that is able to handle ideals of degree over 60000. Such an implementation outperforms the Magma and Singular ones, as shown by our experiments.
First for the shape position case, two methods are designed based on the Wiedemann algorithm: the first is probabilistic and its complexity to complete the change of ordering is O(D(N1+nlog(D)2)), where N1 is the number of nonzero entries of a multiplication matrix; the other is deterministic and computes the LEX Gröbner basis of I via Chinese Remainder Theorem. Then for the general case, the designed method is characterized by the Berlekamp–Massey–Sakata algorithm from Coding Theory to handle the multi-dimensional linearly recurring relations. Complexity analyses of all proposed methods are also provided.
Furthermore, for generic polynomial systems, we present an explicit formula for the estimation of the sparsity of one main multiplication matrix, and prove that its construction is free. With the asymptotic analysis of such sparsity, we are able to show that for generic systems the complexity above becomes O(6/nπD2+n−1n). |
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| AbstractList | Given a zero-dimensional ideal $I \subset \kx$ of degree $D$, the transformation of the ordering of its \grobner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering. The main contributions of this paper are several efficient methods for the change of ordering which take advantage of the sparsity of multiplication matrices in the classical {\sf FGLM} algorithm. Combing all these methods, we propose a deterministic top-level algorithm that automatically detects which method to use depending on the input. As a by-product, we have a fast implementation that is able to handle ideals of degree over $40000$. Such an implementation outperforms the {\sf Magma} and {\sf Singular} ones, as shown by our experiments. First for the shape position case, two methods are designed based on the Wiedemann algorithm: the first is probabilistic and its complexity to complete the change of ordering is $O(D(N_1+n\log (D)))$, where $N_1$ is the number of nonzero entries of a multiplication matrix; the other is deterministic and computes the LEX \grobner basis of $\sqrt{I}$ via Chinese Remainder Theorem. Then for the general case, the designed method is characterized by the Berlekamp--Massey--Sakata algorithm from Coding Theory to handle the multi-dimensional linearly recurring relations. Complexity analyses of all proposed methods are also provided. Furthermore, for generic polynomial systems, we present an explicit formula for the estimation of the sparsity of one main multiplication matrix, and prove its construction is free. With the asymptotic analysis of such sparsity, we are able to show for generic systems the complexity above becomes $O(\sqrt{6/n \pi} D^{2+\frac{n-1}{n}})$. Given a zero-dimensional ideal I⊂K[x1,…,xn] of degree D, the transformation of the ordering of its Gröbner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering. The main contributions of this paper are several efficient methods for the change of ordering which take advantage of the sparsity of multiplication matrices in the classical FGLM algorithm. Combining all these methods, we propose a deterministic top-level algorithm that automatically detects which method to use depending on the input. As a by-product, we have a fast implementation that is able to handle ideals of degree over 60000. Such an implementation outperforms the Magma and Singular ones, as shown by our experiments. First for the shape position case, two methods are designed based on the Wiedemann algorithm: the first is probabilistic and its complexity to complete the change of ordering is O(D(N1+nlog(D)2)), where N1 is the number of nonzero entries of a multiplication matrix; the other is deterministic and computes the LEX Gröbner basis of I via Chinese Remainder Theorem. Then for the general case, the designed method is characterized by the Berlekamp–Massey–Sakata algorithm from Coding Theory to handle the multi-dimensional linearly recurring relations. Complexity analyses of all proposed methods are also provided. Furthermore, for generic polynomial systems, we present an explicit formula for the estimation of the sparsity of one main multiplication matrix, and prove that its construction is free. With the asymptotic analysis of such sparsity, we are able to show that for generic systems the complexity above becomes O(6/nπD2+n−1n). |
| Author | Mou, Chenqi Faugère, Jean-Charles |
| Author_xml | – sequence: 1 givenname: Jean-Charles surname: Faugère fullname: Faugère, Jean-Charles email: Jean-Charles.Faugere@inria.fr organization: Sorbonne Universités, UPMC Univ Paris 06, CNRS, INRIA, Laboratoire d'Informatique de Paris 6 (LIP6), Équipe PolSys, 4 place Jussieu, 75005 Paris, France – sequence: 2 givenname: Chenqi surname: Mou fullname: Mou, Chenqi email: Chenqi.Mou@buaa.edu.cn organization: Sorbonne Universités, UPMC Univ Paris 06, CNRS, INRIA, Laboratoire d'Informatique de Paris 6 (LIP6), Équipe PolSys, 4 place Jussieu, 75005 Paris, France |
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| Keywords | Gröbner bases Sparse matrix Change of ordering Zero-dimensional ideal BMS algorithm Wiedemann algorithm Zero-dimensional ideals Grobner bases |
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| Snippet | Given a zero-dimensional ideal I⊂K[x1,…,xn] of degree D, the transformation of the ordering of its Gröbner basis from DRL to LEX is a key step in polynomial... Given a zero-dimensional ideal $I \subset \kx$ of degree $D$, the transformation of the ordering of its \grobner basis from DRL to LEX is a key step in... |
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| SubjectTerms | BMS algorithm Change of ordering Computational Complexity Computer Science Data Structures and Algorithms Gröbner bases Sparse matrix Symbolic Computation Wiedemann algorithm Zero-dimensional ideal |
| Title | Sparse FGLM algorithms |
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