The resultant of an unmixed bivariate system. (English) Zbl 1068.14070
Summary: This paper gives an explicit formula for computing the resultant of any sparse unmixed bivariate system with a given support. We construct square matrices whose determinant is exactly the resultant, with no extraneous factors. This is the first time that such matrices have been given for unmixed bivariate systems with arbitrary support. The matrices constructed are of hybrid Sylvester and Bézout type. The results extend previous work by the author by giving a complete combinatorial description of the matrix. We make use of the exterior algebra techniques of D. Eisenbud, G. Fløystad and F.-O. Schreyer [Trans. Am. Math. Soc. 355, No.11, 4397–4426 (2003; Zbl 1063.14021)].
MSC:
| 14Q99 | Computational aspects in algebraic geometry |
| 14P99 | Real algebraic and real-analytic geometry |
| 68W30 | Symbolic computation and algebraic computation |
Citations:
Zbl 1063.14021References:
| [1] | Busé, L., 2002. Determinantal 1qresultant. Available from math.AG/0210096 (preprint); Busé, L., 2002. Determinantal 1qresultant. Available from math.AG/0210096 (preprint) |
| [2] | Busé, L.; Elkadi, M.; Mourrain, B., Resultant over the residual of a complete intersection, J. Pure Appl. Algebra, 164, 1-2, 35-57 (2001), Effective methods in algebraic geometry (Bath, 2000) · Zbl 1075.14525 |
| [3] | Cattani, E.; Cox, D.; Dickenstein, A., Residues in toric varieties, Compositio Math., 108, 1, 35-76 (1997) · Zbl 0883.14029 |
| [4] | Chionh, E.-W., Rectangular corner cutting and Dixon \(A\)-resultants, J. Symbolic Comput., 31, 6, 651-669 (2001) · Zbl 0987.65044 |
| [5] | Cox, D., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 4, 1, 17-50 (1995) · Zbl 0846.14032 |
| [6] | Cox, D., Toric residues, Ark. Mat., 34, 1, 73-96 (1996) · Zbl 0904.14029 |
| [7] | Cox, D.; Little, J.; O’Shea, D., Using Algebraic Geometry (1998), Springer-Verlag: Springer-Verlag New York-Berlin-Heidelberg · Zbl 0920.13026 |
| [8] | D’Andrea, C., Macaulay style formulas for sparse resultants, Trans. Amer. Math. Soc., 354, 7, 2595-2629 (2002), (electronic) · Zbl 0987.13019 |
| [9] | D’Andrea, C.; Dickenstein, A., Explicit formulas for the multivariate resultant, J. Pure Appl. Algebra, 164, 1-2, 59-86 (2001) · Zbl 1066.14061 |
| [10] | D’Andrea, C.; Emiris, I., Hybrid sparse resultant matrices for bivariate polynomials, J. Symbolic Comput., 33, 5, 587-608 (2002) · Zbl 1021.65024 |
| [11] | Dickenstein, A.; Emiris, I., Multihomogeneous resultant matrices, (Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation (2002), ACM: ACM New York), 46-54 · Zbl 1072.68660 |
| [12] | Eisenbud, D., Fløystad, G., Schreyer, F.-O., 2000. Sheaf cohomology and free resolutions over exterior algebras. Available from math.AG/0104203 (preprint); Eisenbud, D., Fløystad, G., Schreyer, F.-O., 2000. Sheaf cohomology and free resolutions over exterior algebras. Available from math.AG/0104203 (preprint) · Zbl 1063.14021 |
| [13] | Eisenbud, D., Schreyer, F.-O., 2001. Resultants and Chow forms via exterior syzygies. Available from math.AG/0111040 (preprint); Eisenbud, D., Schreyer, F.-O., 2001. Resultants and Chow forms via exterior syzygies. Available from math.AG/0111040 (preprint) |
| [14] | Emiris, I., 1994. Sparse elimination and applications in kinematics. Ph.D. Thesis. Department of Computer Science, University of California, Berkeley; Emiris, I., 1994. Sparse elimination and applications in kinematics. Ph.D. Thesis. Department of Computer Science, University of California, Berkeley |
| [15] | Emiris, I. Z.; Mourrain, B., Matrices in elimination theory, J. Symbolic Comput., 28, 1-2, 3-44 (1999), Polynomial elimination—algorithms and applications · Zbl 0943.13005 |
| [16] | Fulton, W., Introduction to Toric Varieties (1993), Princeton University Press: Princeton University Press Princeton, NJ, The William H. Roever Lectures in Geometry · Zbl 0813.14039 |
| [17] | Gelfand, I.; Kapranov, M.; Zelevinsky, A., Discriminants, Resultants, and Multidimensional Determinants (1994), Birkhäuser: Birkhäuser Boston-Basel-Berlin · Zbl 0827.14036 |
| [18] | Jouanolou, J.-P., Résultant anisotrope, compléments et applications, Electron. J. Combin., 3, 2 (1996), Research Paper 2, approx. 91 pp. (electronic). The Foata Festschrift · Zbl 0863.13002 |
| [19] | Jouanolou, J.-P., Formes d’inertie et résultant: un formulaire, Adv. Mathe., 126, 2, 119-250 (1997) · Zbl 0882.13008 |
| [20] | Khetan, A., Determinantal formula for the Chow form of a toric surface, (Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation (2002), ACM: ACM New York), 145-150 · Zbl 1072.68677 |
| [21] | Koiran, P., Randomized and deterministic algorithms for the dimension of algebraic varieties, (Proceedings of the 1997 IEEE Conference on Foundations of Computer Science (1997), IEEE), 36-45 |
| [22] | Manocha, D.; Canny, J. F., Real time inverse kinematics of general 6R manipulators, (Proceedings of 1992 IEEE Conference on Robotics and Automation (1992)), 383-389 |
| [23] | Manocha, D.; Canny, J. F., Multipolynomial resultant algorithms, J. Symbolic Comput., 15, 2, 99-122 (1993) · Zbl 0778.13023 |
| [24] | Musta \(ţ\) ă, M., Vanishing theorems on toric varieties, Tohoku Math. J. (2), 54, 3, 451-470 (2002) · Zbl 1092.14064 |
| [25] | Rojas, J. M., Toric laminations, sparse generalized characteristic polynomials, and a refinement of Hilbert’s tenth problem, (Foundations of Computational Mathematics (Rio de Janeiro, 1997) (1997), Springer: Springer Berlin), 369-381 · Zbl 0911.13012 |
| [26] | Rojas, J.M., 2000. Computing complex dimension faster and deterministically. Available from math.AG/0005028 (preprint); Rojas, J.M., 2000. Computing complex dimension faster and deterministically. Available from math.AG/0005028 (preprint) |
| [27] | Weyman, J.; Zelevinski, A., Determinantal formulas for multigraded resultants, J. Algebraic Geom., 3, 569-597 (1994) · Zbl 0816.13007 |
| [28] | Zhang, M., Goldman, R., 2000. Rectangular corner cutting and Sylvester \(A\); Zhang, M., Goldman, R., 2000. Rectangular corner cutting and Sylvester \(A\) · Zbl 1326.68370 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
