Abstract
This paper reexamines univariate reduction from a toric geometric point of view. We begin by constructing a binomial variant of the u-resultant and then retailor the generalized characteristic polynomial to sparse polynomial systems. We thus obtain a fast new algorithm for univariate reduction and a better understanding of the underlying projections. As a corollary, we show that a refinement of Hilbert’s Tenth Problem is decidable in single-exponential time. We also obtain interesting new algebraic identities for the sparse resultant and certain multisymmetric functions.
This research was completed at City University of Hong Kong and was funded by an N.S.F. Mathematical Sciences Postdoctoral Fellowship.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Basu, S., Pollack, R., and Roy, M.F., “On the Combinatorial and Algebraic Complexity of Quantifier Elimination”, In Proc. IEEE Symp. Foundations of Comp. Sci., Santa Fe, New Mexico, 1994, to appear.
Canny, John F., “Generalised Characteristic Polynomials”, J. Symbolic Computation (1990) 9, pp. 241 – 250.
Cornell, Gary and Silverman, Joseph H.,Arithmetic Geometry", with contributions by M. Artin, et. al., Springer-Verlag, 1986.
Danilov, V. I., “The Geometry of Toric Varieties,” Russian Mathematical Surveys, 33 (2), pp. 97 – 154, 1978.
Dyer, M., Gritzmann, P., and Hufnagel, A., “On the Complexity of Computing Mixed Volumes,” SIAM J. Comput., to appear (1996).
Dalbec, John, and Sturmfels, Bernd, Bernd, “Introduction to Chow Forms,” Invariant Methods in Discrete and Computational Geometry (Curaçao, 1994), pp. 37–58, Kluwer Academic Publishers, Dordrecht, 1995.
Emiris, Ioannis Z. and Canny, John F., “Efficient Incremental Algorithms for the Sparse Resultant and the Mixed Volume” Journal of Symbolic Computation, vol. 20 (1995), pp. 117 – 149.
Emiris, Ioannis Z., “Sparse Elimination and Applications in Kinematics,” Ph.D. dissertation, Computer Science Division, U. C. Berkeley (December, 1994), available on-line at http://www. inria. fr/safir/SAFIR/Ioannis. html.
Emiris, Ioannis Z., “On the Complexity of Sparse Elimination,” manuscript, INRIA, March 1996, available online at http://www. inria. fr/safir/SAFIR/Ioannis. html.
Fulton, William,Introduction to Toric Varieties, Annals of Mathematics Studies, no. 131, Princeton University Press, Princeton, New Jersey, 1993.
Giusti, M., Heintz, J., Morais, J. E., Pardo, L. M.,When Polynomial Systems Can Be “Solved” Fast?, Proc. 11thInternational Symposium, AAECC-11, Paris, France, July 17–22, 1995, G. Cohen, M. Giusti and T. Mora, eds., Springer LNCS 948 (1995) pp. 205–231.
Gel’fand, I. M., Kapranov, M. M., and Zelevinsky, A. V., “Discriminants of Polynomials in Several Variables and Triangulations of Newton Polytopes” Algebra and Analysis (translated from Russian) 2, pp. 1 – 62, 1990.
Gel’fand, I. M., Kapranov, M. M., and Zelevinsky, A. V.,Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, 1994.
Gonzalez, L., Lombardi, H, Recio, T., Roy, M.F., “Sturm-Habicht Sequence, Determinants and Real Roots of Univariate Polynomials,” Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation, B. Caviness and J. Johnson, Eds., Springer-Verlag, Wien, New York, 1994, to appear.
Huber, Birkett and Sturmfels, Bernd, Bernd, “Bemshtein’s Theorem in Affine Space” Discrete and Computational Geometry, to appear, 1996.
Khovanskii, A. G., “Newton Polyhedra and Toroidal Varieties,” Functional Anal. Appl., 11 (1977), pp. 289 – 296.
Khovanskii, A. G., “Newton Polyhedra and the Genus of Complete Intersections,” Functional Analysis (translated from Russian), Vol. 12, No. 1, January-March (1978), pp. 51 – 61.
Kapur, Deepak and Lakshman, Yagati N., “Elimination Methods: an Introduction,” Symbolic and Numerical Computation for Artificial Intelligence, B. Donald et. al. (eds. ), Academic Press, 1992.
Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.,Toroidal Embed-dings I, Lecture Notes in Mathematics 339, Springer-Verlag, 1973.
Kapranov, M. M., Sturmfels, B., and Zelevinsky, A. V., “Chow Polytopes and General Resultants,” Duke Mathematical Journal, V. l. 67, No. 1, July, 1992, pp. 189 – 218.
Kushnirenko, A. G., “A Newton Polytope and the Number of Solutions of a System of k Equations in k Unknowns,” Usp. Matem. Nauk., 30, No. 2, pp. 266 – 267 (1975).
Lazard, D. “Résolution des Systèmes d’équations Algébriques,” Theor. Comp. Sci. 15, pp. 146–156.
Lenstra, A. K., Lenstra, H. W., and Lovász, L. “Factoring Polynomials with Rational Coefficients,” Math. Ann. 261 (1982), pp. 515 – 534.
Matiyasevich, Yuri V.Hilbert’s Tenth Problem, MIT Press, MIT, Cambridge, Massachusetts, 1993.
Sturmfels, Bernd, Math 250B (Commutative Algebra), graduate course, U. C. Berkeley, fall 1995, homework #2, problem #3.
Par do-Vasallo, Luis-Miguel and Krick, Teresa, “A Computational Method for Diophantine Approximation,” to appear in Proc. MEGA’94, Birkhäuser, Progress in Math. (1995).
Pedersen, P. and Sturmfels, B., “Product Formulas for Sparse Resultants and Chow Forms,” Mathematische Zeitschrift, 214: 377 – 396, 1993.
Renegar, Jim, “On the Worst Case Arithmetic Complexity of Approximating Zeros of Systems of Polynomials,” Technical Report, School of Operations Research and Industrial Engineering, Cornell University.
Rojas, J. Maurice, “A Convex Geometric Approach to Counting the Roots of a Polynomial System,” Theoretical Computer Science (1994), vol. 133 (1), pp. 105–140.
Rojas, J. Maurice “Toric Intersection Theory for Affine Root Counting,” submitted to the Journal of Pure and Applied Algebra, also available on-line at http://www-math. mit. edu/~rojas.
Rojas, J. Maurice “When do Resultants Really Vanish?,” manuscript, MIT.
Rojas, J. Maurice “Affine Elimination Theory and Solving Certain Diophantine Systems Quickly,” manuscript, MIT.
Roy, Marie-Prangoise “Basic Algorithms in Real Algebraic Geometry and their Complexity: from Sturm Theorem to the Existential Theory of Reals,” manuscript, IRMAR, Rennes, Prance, 1995.
Rojas, J. M., and Wang, Xiaoshen, “Counting Affine Roots of Polynomial Systems Via Pointed Newton Polytopes,” Journal of Complexity, vol. 12, June (1996), pp. 116–133, also available on-line at http://www-math. mit. edu/~rojas.
Shub, Mike, Mike, "Some Remarks on Bézout’s Theorem and Complexity Theory, Prom Topology to Computation: Proceedings of the Smalefest, pp. 443–455, Springer-Verlag, 1993.
Sturmfels, Bernd, “Sparse Elimination Theory,” In D. Eisenbud and L. Rob-biano, editors, Proc. Computat. Algebraic Geom. and Commut. Algebra 1991, pages 377–396, Cortona, Italy, 1993, Cambridge Univ. Press.
Sturmfels, Bernd, “On the Newton Polytope of the Resultant,” Journal of Algebraic Combinatorics, 3: 207 – 236, 1994.
Sturmfels, Bernd,Gröbner Bases and Convex Polytopes, Lectures presented at the Holiday Symposium at New Mexico State University, December 27 – 31, 1994.
Ziegler, Gunter M.,Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995.
Zippel, R.,Effective Polynomial Computation, Kluwer Academic Publishers, Boston, 1993.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Rojas, J.M. (1997). Toric Laminations, Sparse Generalized Characteristic Polynomials, and a Refinement of Hubert’s Tenth Problem. In: Cucker, F., Shub, M. (eds) Foundations of Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60539-0_30
Download citation
DOI: https://doi.org/10.1007/978-3-642-60539-0_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61647-4
Online ISBN: 978-3-642-60539-0
eBook Packages: Springer Book Archive