## The Structure of Sparse Resultant Matrices (1997)

Venue: | In Proc. ACM Intern. Symp. on Symbolic and Algebraic Computation |

Citations: | 22 - 11 self |

### BibTeX

@INPROCEEDINGS{Emiris97thestructure,

author = {Ioannis Z. Emiris and Victor Y. Pan},

title = {The Structure of Sparse Resultant Matrices},

booktitle = {In Proc. ACM Intern. Symp. on Symbolic and Algebraic Computation},

year = {1997},

pages = {18919--6},

publisher = {ACM Press}

}

### OpenURL

### Abstract

Resultants characterize the existence of roots of systems of multivariate nonlinear polynomial equations, while their matrices reduce the computation of all common zeros to a problem in linear algebra. Sparse elimination theory has introduced the sparse resultant, which takes into account the sparse structure of the polynomials. The construction of sparse resultant, or Newton, matrices is a critical step in the computation of the resultant and the solution of the system. We exploit the matrix structure and decrease the time complexity of constructing such matrices to roughly quadratic in the matrix dimension, whereas the previous methods had cubic complexity. The space complexity is also decreased by one order of magnitude. These results imply similar improvements in the complexity of computing the resultant itself and of solving zero-dimensional systems. We apply some novel techniques for determining the rank of rectangular matrices by an exact or numerical computation. Finally, we im...

### Citations

163 |
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(Show Context)
Citation Context ...rather large inputs, due to its high constant factor. Our methods can be adapted to other basic algorithms for polynomial multiplication of intermediate speed, namely the so-called Karatsuba's method =-=[11]-=-, which may be preferable for inputs of moderate size. Karatsuba's multiplication algorithm has linear space complexity and time complexitysO(k lg3 ) for k-degree polynomials, where lg denotes the log... |

88 |
Effective Polynomial Computation
- Zippel
- 1993
(Show Context)
Citation Context ...via the support cardinality or the Newton polytope. The existing general bounds are interesting only in the dense case [1]. Sparse interpolation has received a lot of attention; see the algorithms in =-=[19, 9]-=-, supporting complexity linear in the product of n, the maximum degree in any single variable and a bound on the number of monomials. Section 4 improves these bounds by exploiting the structure of non... |

79 |
On the Newton polytope of the resultant
- Sturmfels
- 1994
(Show Context)
Citation Context ...ster's as well as Macaulay's constructions, have been proposed for constructing Newton, or sparse resultant, matrices: The subdivision-based algorithm of [2] (subsequently improved and generalized in =-=[4, 16]-=-) and the incremental algorithm of [7], which constructs a rectangular matrix and then obtains a square nonsingular submatrix. Canny, Kaltofen and Lakshman [3] studied the structure of Macaulay matric... |

75 | An efficient algorithm for the sparse mixed resultant
- Canny, Emiris
(Show Context)
Citation Context ...on 5). Two main algorithms, generalizing Sylvester's as well as Macaulay's constructions, have been proposed for constructing Newton, or sparse resultant, matrices: The subdivision-based algorithm of =-=[2]-=- (subsequently improved and generalized in [4, 16]) and the incremental algorithm of [7], which constructs a rectangular matrix and then obtains a square nonsingular submatrix. Canny, Kaltofen and Lak... |

53 |
EOEcient incremental algorithms for the sparse resultant and the mixed volume
- Emiris, Canny
- 1996
(Show Context)
Citation Context ..., have been proposed for constructing Newton, or sparse resultant, matrices: The subdivision-based algorithm of [2] (subsequently improved and generalized in [4, 16]) and the incremental algorithm of =-=[7]-=-, which constructs a rectangular matrix and then obtains a square nonsingular submatrix. Canny, Kaltofen and Lakshman [3] studied the structure of Macaulay matrices Our results generalize their approa... |

48 |
Solving systems of non-linear polynomial equations faster
- Canny, Kaltofen, et al.
- 1989
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Citation Context ...olution of polynomial systems. Construction and manipulation of Newton matrices is a critical operation in some of the most eOEcient known algorithms for solving zero-dimensional systems of equations =-=[3, 12, 5, 13]-=-. Our practical motivation is the real-time solution of systems with, say, up to 10 variables; or the computation of the resultant polynomial, for instance in graphics and modeling applications where ... |

45 | Improved sparse multivariate polynomial interpolation algorithms
- Kaltofen, Lakshman
- 1988
(Show Context)
Citation Context ...via the support cardinality or the Newton polytope. The existing general bounds are interesting only in the dense case [1]. Sparse interpolation has received a lot of attention; see the algorithms in =-=[19, 9]-=-, supporting complexity linear in the product of n, the maximum degree in any single variable and a bound on the number of monomials. Section 4 improves these bounds by exploiting the structure of non... |

40 | Solving systems of polynomial equations
- Manocha
- 1994
(Show Context)
Citation Context ...olution of polynomial systems. Construction and manipulation of Newton matrices is a critical operation in some of the most eOEcient known algorithms for solving zero-dimensional systems of equations =-=[3, 12, 5, 13]-=-. Our practical motivation is the real-time solution of systems with, say, up to 10 variables; or the computation of the resultant polynomial, for instance in graphics and modeling applications where ... |

26 | On the complexity of sparse elimination
- Emiris
- 1996
(Show Context)
Citation Context ...olution of polynomial systems. Construction and manipulation of Newton matrices is a critical operation in some of the most eOEcient known algorithms for solving zero-dimensional systems of equations =-=[3, 12, 5, 13]-=-. Our practical motivation is the real-time solution of systems with, say, up to 10 variables; or the computation of the resultant polynomial, for instance in graphics and modeling applications where ... |

25 | Solving special polynomial systems by using structured matrices and algebraic residues
- Mourrain, Pan
- 1997
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Citation Context |

22 | An algorithm for the Newton resultant - Canny, Pedersen - 1993 |

19 | Numerical Computation of a Polynomial GCD and Extensions
- Pan
- 1998
(Show Context)
Citation Context ...exceed a ��xed small positive tolerance value ffl. This can be achieved by means of any black box algorithm for computing the Singular Value Decomposition (SVD) of M . A much less costly algorithm=-= of [14] exp-=-loits the structure of M . The algorithm avoids computing SVD and is rational, that is, only involves ��eld operations, which can also be performed in exact arithmetic. Theorem 3.4 Let C and G be ... |

18 | Parallel computation of polynomial GCD and some related parallel computations over abstract elds - Pan - 1996 |

11 | Fundamental Algorithms
- Bini, Pan
- 1994
(Show Context)
Citation Context ...se multivariate polynomials within the computational complexity bounds expressed via the support cardinality or the Newton polytope. The existing general bounds are interesting only in the dense case =-=[1]-=-. Sparse interpolation has received a lot of attention; see the algorithms in [19, 9], supporting complexity linear in the product of n, the maximum degree in any single variable and a bound on the nu... |

9 | A general solver based on sparse resultants: Numerical issues and kinematic applications
- Emiris
- 1997
(Show Context)
Citation Context ...tests the nonsingularity of several matrix candidates, by applying an incremental version of LU-decomposition, which is performed in place. Table 2 shows the various parameters in examples studied in =-=[7, 6]. Th-=-e ��rst three are multihomogeneous systems with 3 groups of 2, 1 and 1 variables respectively, where the corresponding degrees are given in the table, the following two are dioeerent expressions o... |

8 | Solving sparse linear equations over nite elds - Wiedemann - 1986 |

6 | Techniques for exploiting structure in matrix formulae of the sparse resultant - Emiris, Pan - 1996 |

4 | Processor eOEcient parallel solution of linear systems over an abstract ��eld - Kaltofen, Pan - 1991 |