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Last step in the proof of Lemma 6. The proof is, for the most part, analogous to that of Lemma 6. By Definition 6. The next step is to determine the point corresponding to the toric Jacobian. We shall consider three different cases. This is the unique point claimed. Cell C0 of Q1 ; the vertex chain to the left is where b0 may belong. Cases of the proof of Lemma 6. Vertices ai and cells Ci are denoted a[i] and C[i] respectively.

If points b0 or bg exist defined in Algorithm 5.


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Recall that by Algorithm 5. Example 6. All twofold mixed volumes are equal to 1. Shown are certain cells of the subdivision for each system. For a well-defined matrix, we show that it suffices to consider all points in F. The first case is when p does not belong to a 0-mixed cell, i. By Proposition 5. Hence, the lower bound is always satisfied for sufficiently large M. Point p is not that of Definition 6. If F0 is a two-dimensional cell, say containing ar1 , then by Lemma 6. It remains to consider F0 as an edge. F0 and F1 must be non-parallel edges, so that a mixed cell can be generated.

These form cells, in the mixed subdivision, which are sums of vertices from Q0 , Q1 and one edge of Q2. Hybrid sparse resultant matrices Theorem 6. Main Let M be the matrix constructed at the end of Section 4 with the lifting defined in Section 5. Theorem 6.

The fact that det M is a multiple of the resultant follows from Cattani et al. The determinant has the same degree in the coefficients of f0 as the resultant because the number of rows depending on the coefficients of f0 is exactly MV Q1 , Q2 , and all the rows depend linearly on those coefficients; cf. Cox et al. As the polynomial G f is well defined modulo the homogeneous ideal, it is enough to show that the determinant of M is non-zero for a specific choice of it. This can be done because of Lemma 6. We shall denote by M f p the matrix obtained by eliminating the rows and columns indexed by p in M f.

In all the cases, they are different from zero. The extraneous factor pM is a polynomial in Z[c] with content 1. MV Q1 ,Q2 Proof. Example 7. Let us compute the resultant of the family of Example 3. The Qi are equal to the unitary square. Q2 is not divided under the lifting. Shown are the cells of the Qi , certain cells of the subdivision, and the point p. F has 19 elements. The cells in the Qi , point p, and certain cells of the subdivision are shown in Figure 6. Here is the Maple session, supposing the lifting is known and given as the last argu- ment. Then the mixed volumes become 12, 12 and 6, so the total degree of the sparse resultant is The corresponding Maple session is the following, using the variables from the session above.

Consider the system of Figure 7. Q1 , Q2 are each subdivided into three linearly lifted cells. The vertex liftings which are not explicit can be deduced by the liftings of three points in the same cell. This leaves the point 6, 10 , circled in the figure, corresponding to f0 s.

The supports of G f and toric Jacobian J f have cardinalities 5 and Table 1 starts with homogeneous systems of degree d. The next row regards surface implicitization from Manocha, , p.


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We took the union of these supports as W , then specialized the missing coefficients to zero. The last rows correspond to examples discussed in this paper. The two shaded cells are those from Q0. Hybrid sparse resultant matrices Table 1. Comparison of resultant matrices. In some of these algorithms, it is necessary to fill the supports with zero coefficients in order to satisfy the hypotheses; in this case, there is no guarantee that the matrix determinant is nonzero. We plan further applications on scaled Newton polygons that should emphasize the merits of our construction.

An important parameter in comparisons is the degree of the matrix determinant. For all methods yielding square matrices, this equals the dimension of matrix M. For our method, as well as Cattani et al. For bihomogeneous systems, an optimal sparse resultant matrix exists. Acknowledgements We thank R.

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Goldman for specifying the corner-cutting method for d-homogeneous systems and the anonymous referees for their suggestions. References Aries, F, Senoussi, R. An implicitization algorithm for rational surfaces with no base points. Canny, J.

May A subdivision-based algorithm for the sparse resultant. ACM 47, 3, — An algorithm for the Newton resultant. In Tech.

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Science Dept, Cornell University. Cattani, E. Residues in toric varieties. Residues and resultants. Tokyo, 5, — Cox, D. Macaulay-style formulas for the sparse resultant. AMS, to appear. Zelevinsky , Discriminants, resultants, and multidimensional determinants, Math 90 , no. Macaulay , Some formulae in elimination, Proc. London Math. Buse inria.

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Version This documentation describes version 1. Source code The source code from which this documentation is derived is in the file EliminationMatrices. The choice of the minors is according to the construction of the determinant of a complex regularityVar -- computes the Castelnuovo-Mumford regularity of homogeneous ideals in terms of Betti numbers, with respect to some of the variables of the ring. Symbols byResolution -- Strategy for eliminationMatrix. CM2Residual -- Strategy for eliminationMatrix.

Exact -- Strategy for functions that uses rank computation. EN to discriminate to distinguish.

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FR discriminant. A stepwise discriminant analysis of delinquent and nondelinquent youth. The Council should not act selectively or discriminate between regions and situations. A stepwise discriminant analysis of delinquent and nondelinquent youth , dans Violato, C. Synonyms Synonyms French for "discriminant":. Context sentences Context sentences for "discriminant" in English These sentences come from external sources and may not be accurate.