## A Subdivision-Based Algorithm for the Sparse Resultant (1999)

Venue: | J. ACM |

Citations: | 34 - 8 self |

### BibTeX

@ARTICLE{Canny99asubdivision-based,

author = {John F. Canny and Ioannis Z. Emiris},

title = {A Subdivision-Based Algorithm for the Sparse Resultant},

journal = {J. ACM},

year = {1999},

volume = {47},

pages = {417--451}

}

### Years of Citing Articles

### OpenURL

### Abstract

Multivariate resultants generalize the Sylvester resultant of two polynomials and characterize the solvability of a polynomial system. They also reduce the computation of all common roots to a problem in linear algebra.

### Citations

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Citation Context ...he next theorem, by combining the two previous lemmas. If an explicit dense representation is required, then an additional cost of O(jEj 2 ) must be included. For di erent matrix representations, see =-=[AHU74]-=-. Theorem 11.6 Given polynomials f1;:::;fn+1, the algorithm computes an implicit representation of Newton matrix M with worst-case bit complexity O jEjn 9:5 6:5 log 2 2 d log 1 l where is the maximum ... |

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Citation Context ...polytopes Q1;:::;Qn+1 and the pair RC(p) 2 f1; 2;:::;n +1g N at every p 2E. 9sDescription The rst stage constructs the vertex sets of Qi by repeated application of linear programming, as discussed in =-=[GLS93]-=-. For a point aik in Ai = fai1;:::;ai ig, we test whether there is a feasible solution to the following system. aik = j X j=1;j6=k jaij; j X j=1;j6=k j =1; j 0; j =1;:::; i: Feasibility implies aik is... |

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Citation Context ... have ignored polylogarithmic factors, and e denotes the exponential base. This section applies the current record asymptotic complexity for matrix multiplication, namely O(N 2:376 ) for N N matrices =-=[CW90]-=-. In manipulating univariate polynomials, we shall use asymptotic bounds based on the Fast Fourier Transform (FFT). The running time for the interpolation of a univariate polynomial of degree D from i... |

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Citation Context ...limination and some necessary de nitions for the study of polynomial systems. For background on polynomial systems see [vdW50, Zip93], and for background on combinatorial geometry and polytope theory =-=[Sch93]-=-. Sparse elimination theory generalizes several results of classical elimination theory on multivariate polynomial systems by considering the structure of the given polynomials, namely their Newton po... |

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Citation Context ...l Newton polytopes may be reduced to linear programming. We can apply any polynomial-time algorithm such as Khachiyan's ellipsoid method or Karmarkar's algorithm. In the case of Karmarkar's algorithm =-=[Kar84]-=- the bit complexity for a linear program of V variables, C constraints and B bits per coe cient is O (C 2 V 5:5 B 2 ): Let d be the highest degree of any input polynomial in a single variable and = ma... |

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Citation Context ... a resultant matrix de ned generically. The naive GCD approach can be used after a suitable perturbation of the specialized system, but here we propose two economical methods based on the approach of =-=[Can88]-=-. 9.1 Division Method Let g1;:::;gn+1 be the specializations of the given polynomials with coe cients in an arbitrary coe cient eld. The idea is to introduce some indeterminates that will guarantee th... |

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Citation Context ...1 Ai is n-dimensional. This technical hypothesis is removed in [CP93]. This lattice is identi ed with Zn possibly after a change of variables, which can be implemented by applying Smith's normal form =-=[Stu94]-=-. Under this hypothesis the resultant's degree can be speci ed. Proposition 3.7 [PS93] The sparse resultant is separately homogeneous in the coe cients (ci1;:::;ci ) of each i fi and its degree in the... |

77 | An efficient algorithm for the sparse mixed resultant
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Citation Context ...ly smaller and often exact. Our main contribution is to present historically the rst general and eOEcient algorithm to compute the sparse resultant, by completing and expanding on the construction of =-=[CE93]-=-. The algorithm denes a matrix with entries equal to the input coeOEcients or zero, whose determinant is a nontrivial multiple of the sparse resultant, Computer Science Division, University of Califor... |

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Citation Context ...m of dimensions of all b Fi, for 1 i n +1, is larger or equal to the face dimension, which lies in f0; 1;:::;ng. The genericity requirement on li implies the following property; the proof is found in =-=[BS92]-=-. n+1 dim F1 b + + dim Fn+1 b X = dim Moreover, given a face in b , the collection b F1;:::; b Fn+1 is uniquely de ned by ,1 jQ. The image of b under induces a polyhedral subdivision of Q whose maxima... |

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Citation Context ...e positive volume. We nowexamine the complexity of constructing M. The change of variables that may be required to ensure that Pn+1 i=1 Ai generates Zn involves the computation of a Smith normal form =-=[HM91]-=-. Its asymptotic complexity is dominated so we ignore it in the sequel. Identifying the vertices of all Newton polytopes may be reduced to linear programming. We can apply any polynomial-time algorith... |

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EÆcient incremental algorithms for the sparse resultant and the mixed volume
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Citation Context ...mized algorithm has been proposed by Emiris and Canny in order to construct Newton matrices of smaller size, which are optimal for all classes of polynomial systems where such matrices provably exist =-=[EC95]-=-. More recent work focuses on the structure of these matrices, which has been shown to generalize Toeplitz structure [EP97]. For solving systems of n polynomial equations in n unknowns, certain proper... |

53 |
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Citation Context ...zero set) of f1;:::;fn. This construction is too costly to have any practical signi cance. The rst constructive methods for computing and evaluating the sparse resultant were proposed by Sturmfels in =-=[Stu93]-=-, the most e cient having complexity super-polynomial in the total degree of the sparse resultant and exponential in n with a quadratic exponent. A greedy variant of our algorithm has been proposed by... |

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Citation Context ...ubic surface, methods based on customized resultants exploiting sparseness have achieved a speedup of at least 10 3 in the running time over approaches relying on Gröbner bases and the Ritt-Wu method=-= [MC93].-=- Sparseness can lead to an important improvement over classical elimination methods in practice. To illustrate the disparitybetween the classical Bézout bound and Bernstein's sparse bound, consider t... |

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Citation Context ...1980's in the work of Gelfand, Kapranov and Zelevinsky [GKZ94]. The sparse resultant was called the (A1;:::;An+1)-resultant, where Ai Z n is the support of the i-th polynomial. Pedersen and Sturmfels =-=[PS93]-=-gave a product formula of Poisson type for the sparse resultant, namely R 0 Q 2V (f1;:::;fn) fn+1( ), where the extraneous factor R 0 is a rational function in the coe cients of f1;:::;fn and V (f1;::... |

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Citation Context ...the notion of sparseness in section 3. In general, the sparse resultant has smaller degree than its classical counterpart because its degree depends on the Bernstein bound on the number of a ne roots =-=[Ber75].-=- Bernstein's bound is at most equal to Bézout's and, for sparse systems, it is substantially smaller and often exact. Our main contribution is to present historically the rst general and e cient algo... |

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Citation Context ...the total degree of the sparse resultant. Thus the GCD computation is branch-free and reduces to calculation of the appropriate minors of the Sylvester matrix, or subresultants, of D1(u) and D0 1 (u) =-=[Loo82]-=-. The complexity of the method is analyzed in section 12.2. Once again, there is a more favorable case that simpli es the situation. Namely, if the given polynomials gi are generic enough, meaning tha... |

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Citation Context ... properties of the Newton matrix must be established. One approach is that of [PS96], whereas a simpler proof, based on the present algorithm, which reduces root- nding to an eigenproblem is found in =-=[Emi96]-=-. This result extends older work in [AS88, MC93]. Observe that the exact sparse resultant is not necessary, since a Newton matrix su ces. The computation of solution sets with positive dimension has b... |

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Citation Context ...olytopes lie in a two-dimensional plane, as shown in gure 4 from a perspective view. The gure also depicts the lower envelope s(Q) of their Minkowski sum b Q constructed by the convex hull program of =-=[Emi98]-=-. The program automatically triangulates all output facets; this triangulation is immaterial in inducing a mixed subdivision of Q. This gure, as well as gure 5, are drawn by Geomview. Figure 5 depicts... |

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Citation Context ...mal for all classes of polynomial systems where such matrices provably exist [EC95]. More recent work focuses on the structure of these matrices, which has been shown to generalize Toeplitz structure =-=[EP97]-=-. For solving systems of n polynomial equations in n unknowns, certain properties of the Newton matrix must be established. One approach is that of [PS96], whereas a simpler proof, based on the presen... |

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Citation Context ...all matrices whose entries are polynomials in the input coe cients. Its generalization, the Bezoutian matrix, works in arbitrary dimension and does not require any genericity assumption on the inputs =-=[Mou97]-=-. Another generalization is Dixon's matrix; see, for instance, [KS95]. A di erent approach that also obtains determinantal expressions for the resultant stems from Sylvester's formulation for n =2. Th... |

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An Algorithm for the Newton Resultant
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Citation Context ...aving complexity super-polynomial in the total degree of the sparse resultant and exponential in n with a quadratic exponent. A greedy variant of our algorithm has been proposed by Canny and Pedersen =-=[CP93]-=- which, based on a mixed subdivision, typically constructs smaller matrices. This algorithm also removes a rather technical requirement on the input supports, formalized in section 3. A second general... |

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Citation Context ...s generalization, the Bezoutian matrix, works in arbitrary dimension and does not require any genericity assumption on the inputs [Mou97]. Another generalization is Dixon's matrix; see, for instance, =-=[KS95]-=-. A di erent approach that also obtains determinantal expressions for the resultant stems from Sylvester's formulation for n =2. This is the determinant of a matrix of minimum possible size, where the... |

17 |
An e cient algorithm for the sparse mixed resultant
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- 1993
(Show Context)
Citation Context ...lly smaller and often exact. Our main contribution is to present historically the rst general and e cient algorithm to compute the sparse resultant, by completing and expanding on the construction of =-=[CE93]-=-. The algorithm de nes a matrix with entries equal to the input coe cients or zero, whose determinant isanontrivial multiple of the sparse resultant, Computer Science Division, University of Californi... |

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Citation Context ...nds older work in [AS88, MC93]. Observe that the exact sparse resultant is not necessary, since a Newton matrix su ces. The computation of solution sets with positive dimension has been undertaken in =-=[KM95]-=-. In practice, questions of degeneracy may have to be addressed, as in [Man94, Mou98, Roj99]. A related problem of independent signi cance is the computation of mixed volumes. Di erent algorithms and ... |

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Citation Context ... Qn+1 and the pair RC(p) 2 f1; 2; : : : ; n + 1g \Theta N at every p 2 E . 9 Description The rst stage constructs the vertex sets of Q i by repeated application of linear programming, as discussed in =-=[GLS93]-=-. For a point a ik in A i = fa i1 ; : : : ; a i i g, we test whether there is a feasible solution to the following system. a ik =sj X j=1;j 6=ksj a ij ;sj X j=1;j 6=ksj = 1;sjs0; j = 1; : : : ;si : Fe... |

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Citation Context ...ime and the accuracy compare favorably to sparse homotopies exploiting the monomial structure; furthermore, the accuracy is higher than that of least-square techniques, which require more information =-=[Emi97]-=-. The remainder of this article is organized as follows. The next section points to previous work and compares it with ours. Section 3 introduces the basic notions of sparse elimination theory, and pr... |

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Citation Context ...l has volume at least 1=n! because it is de ned by integer points. The volume of a convex polytope is asymptotically equal to the number of integer lattice points in its interior, by Ehrart's theorem =-=[Ehr67]-=-. The number of maximal cells is then O(jEj n!). Since every (n , 1)-dimensional cell can be de ned as the intersection of two maximal cells, the number of the former is at most O((jEj n!) 2 ). We use... |