Abstract
In this paper we present an elimination method for algebraically closed fields based on Seidenberg’s theory. The method produces, for any pair [PS, QS] of sets of multivariate polynomials, a sequence of triangular forms TF 1,…, TF e and polynomial sets US 1,…, US e such that the difference set of common zeros of PS and QS is the same as the union of the difference sets of common zeros of TF i and US i. Moreover, the triangular forms TF i and polynomial sets US i can be so computed as to give a necessary and sufficient condition for the given system to have algebraic zeros for some prescribed variables. This method has a number of applications such as to solving systems of polynomial equations and inequalities, mechanical theorem proving in geometry, irreducible decomposition of algebraic varieties and constructive proof of Hilbert’s Nullstellensatz which are partially discussed in the paper. Preliminary experiments show that the efficiency of this method is at least comparable with that of the well-known methods of characteristic sets and Gröbner bases for some applications. A few illustrative yet encouraging examples performed by a draft implementation of the method are given.
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
B. Buchberger, An Algorithmical Criterion for the Solvability of Algebraic Systems of Equations (in German), Aequantiones Math. 4, 374–383 (1970).
B. Buchberger, Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory, Multidimensional Systems Theory (N. K. Bose, ed.), D. Reidel Publishing Company, Dordrecht-Boston, 184–232 (1985).
J. F. Canny, E. Kaltofen and L. Yagati, Solving Systems of Non-Linear Polynomial Equations Faster, Proc. ISSAC’89 (G. H. Gonnet, ed.), 121–128 (1989).
G. Carrà Ferro and G. Gallo, A Procedure to Prove Geometrical Statements, Proc. AAECC-5, Lecture Notes in Comput. Sci. 356, 141–150 (1987).
D. Yu. Grigoriev and A. L. Chistiv, Subexponential-Time Solving Systems of Algebraic Equations, Preprints LOMI E-9-83 and E-10-83, Leningrad (1983).
D. Yu. Grigoriev and A. L. Chistiv, Fast Decomposition of Polynomials into Irreducible Ones and the Solution of Systems of Algebraic Equations, Soviet Math. Dokl. (AMS Translation) 29, 380–383 (1984).
W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Vol. I, II, Cambridge University Press, Cambridge (1947/1952).
M. Kalkbrener, Three Contributions to Elimination Theory, Ph.D thesis, RISC-LINZ, Johannes Kepler University, Austria (1991).
D. Kapur, Using Gröbner Bases to Reason About Geometry Problems, J. Symb. Comput. 2, 399–408 (1996).
D. Lazard, Systems of Algebraic Equations, Proc. EUROSAM’79 (E. W. Ng, ed.), 88–94 (1979).
D. Lazard, Resolution des systemes d’equation algebriques, Theoretical Comput. Sci. 15, 77–110 (1981).
D. Lazard, A New Method for Solving Algebraic Systems of Positive Dimension, Discrete Applied Math. 32, to appear (1991).
D. Lazard, Solving Zero-dimensional Algebraic Systems, J. Symb. Comput. 13, 117–131 (1992).
F. S. MaCaulay, The Algebraic Theory of Modular Systems, Stechert-Hafner Service Agency, New York-London (1964).
J. F. Ritt, Differential Equations from the Algebraic Standpoint, Amer. Math. Soc., New York (1932).
J. F. Ritt, Differential Algebra, Amer. Math. Soc., New York (1950).
A. Seidenberg, Some Remarks on Hilbert’s Nullstellensatz, Arch. Math. 7, 235–240 (1956).
A. Seidenberg, An Elimination Theory for Differential Algebra, Univ. California Publ. Math. (N.S.) 3(2), 31–66 (1956).
A. Seidenberg, On k-Constructable Sets, k-Elementary Formulae, and Elimination Theory, J. reine angew. Math. 239/240, 256–267 (1969).
B. Sturmfels, Sparse Elimination Theory, Presented at the Meeting on Comput. Algebraic Geometry and Commutative Algebra (Cortona, June 1991).
D. M. Wang, Characteristic Sets and Zero Structure of Polynomial Sets, Lecture Notes, RISC-LINZ, Johannes Kepler University, Austria (1989).
D. M. Wang, An Implementation of the Characteristic Set Method in Maple, Proc. DISCO’92 (Bath, England, April 13–15, 1992), to appear (1992).
D. M. Wang, On Wu’s Method for Solving Systems of Algebraic Equations, RISC-Linz Series no. 91-52.0, Johannes Kepler University, Austria (1991).
D. M. Wang, Reasoning about Geometric Problems Using Algebraic Methods, MEDLAR 24-Month Deliverables, to appear (1991).
D. M. Wang, A Strategy for Speeding up the Computation of Characteristic Sets, Proc. 17th Int. Symp. Math. Foundations of Comput. Sci. (Prague, August 24–28, 1992), to appear (1992).
D. M. Wang, An Elimination Method for Polynomial Systems, Preprint, RISC-LINZ, Johannes Kepler University, Austria (1992).
W. T. Wu, Basic Principles of Mechanical Theorem Proving in Elementary Geometries, J. Sys. Sci. & Math. Scis. 4, 207–235 (1984); J. Automated Reasoning 2, 221–252 (1986).
W. T. Wu, On Zeros of Algebraic Equations — An application of Ritt principle, Kexue Tongbao 31, 1–5 (1986).
W. T. Wu, A Zero Structure Theorem for Polynomial Equations-Solving, MM Research Preprints, No. 1, 2–12 (1987).
W. T. Wu, On a Projection Theorem of Quasi-Varieties in Elimination Theory, Chinese Ann. Math. Ser. B 11(2), 220–226 (1990).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Birkhäuser Boston
About this paper
Cite this paper
Wang, D. (1993). An Elimination Method Based on Seidenberg’s Theory and Its Applications. In: Eyssette, F., Galligo, A. (eds) Computational Algebraic Geometry. Progress in Mathematics, vol 109. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2752-6_21
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2752-6_21
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7652-4
Online ISBN: 978-1-4612-2752-6
eBook Packages: Springer Book Archive