Abstract
This chapter will study systematic methods for eliminating variables from systems of polynomial equations. The basic strategy of elimination theory will be given in two main theorems: the Elimination Theorem and the Extension Theorem. We will prove these results using Gröbner bases and the classic theory of resultants.
The original version of this chapter was revised. A correction to this chapter is available at https://doi.org/10.1007/978-3-319-16721-3_11
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
References
J. Abbott, A. Bigatti, G. Lagorio, CoCoA-5: A System for Doing Computations in Commutative Algebra (2014), available at http://cocoa.dima.unige.it
W. Adams, P. Loustaunau, An Introduction to Gröbner Bases. Graduate Studies in Mathematics, vol. 3 (AMS, Providence, 1994)
C. Aholt, B. Sturmfels, R. Thomas, A Hilbert scheme in computer vision. Can. J. Math. 65, 961–988 (2013)
D. Anderson, R. Goldman, T. Sederberg, Implicit representation of parametric curves and surfaces. Comput. Vis. Graph. Image Des. 28, 72–84 (1984a)
D. Anderson, R. Goldman, T. Sederberg, Vector elimination: a technique for the implicitization, inversion and intersection of planar parametric rational polynomial curves. Comput. Aided Geom. Des. 1, 327–356 (1984b)
E. Arnold, S. Lucas, L. Taalman, Gröbner basis representations of sudoku. Coll. Math. J. 41, 101–112 (2010)
M. Aschenbrenner, Ideal membership in polynomial rings over the integers. J. Am. Math. Soc. 17, 407–441 (2004)
M.F. Atiyah, I.G. MacDonald, Introduction to Commutative Algebra (Addison-Wesley, Reading, MA, 1969)
P. Aubry, D. Lazard, M. Moreno Maza, On the theories of triangular sets. J. Symb. Comput. 28, 105–124 (1999)
J. Baillieul et al., Robotics. In: Proceedings of Symposia in Applied Mathematics, vol. 41 (American Mathematical Society, Providence, Rhode Island, 1990)
A.A. Ball, The Parametric Representation of Curves and Surfaces Using Rational Polynomial Functions, in The Mathematics of Surfaces, II, ed. by R.R. Martin (Clarendon Press, Oxford, 1987), pp. 39–61
D. Bayer, The division algorithm and the Hilbert scheme, Ph.D. thesis, Harvard University, 1982
D. Bayer, D. Mumford, What Can Be Computed in Algebraic Geometry?, in Computational Algebraic Geometry and Commutative Algebra, ed. by D. Eisenbud, L. Robbiano (Cambridge University Press, Cambridge, 1993), pp. 1–48
D. Bayer, M. Stillman, A criterion for detecting m-regularity. Invent. Math. 87, 1–11 (1987a)
D. Bayer, M. Stillman, A theorem on refining division orders by the reverse lexicographic order. Duke J. Math. 55, 321–328 (1987b)
D. Bayer, M. Stillman, On the Complexity of Computing Syzygies, in ComputationalAspects of Commutative Algebra, ed. by L. Robbiano (Academic Press, New York, 1988), pp. 1–13
T. Becker, V. Weispfenning, Gröbner Bases (Springer, New York-Berlin-Heidelberg, 1993)
C.T. Benson, L.C. Grove, Finite Reflection Groups, 2nd edn. (Springer, New York-Berlin-Heidelberg, 1985)
A. Bigatti, Computation of Hilbert-Poincaré series. J. Pure Appl. Algebra 119, 237–253 (1997)
E. Brieskorn, H. Knörrer, Plane Algebraic Curves (Birkhäuser, Basel-Boston-Stuttgart, 1986)
J.W. Bruce, P.J. Giblin, Curves and Singularities, 2nd edn. (Cambridge University Press, Cambridge, 1992)
B. Buchberger, Ein algorithmus zum auffinden der basiselemente des restklassenrings nach einem nulldimensionalen polynomideal, Doctoral Thesis, Mathematical Institute, University of Innsbruck, 1965. English translation An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal by M.P. Abramson, J. Symb. Comput. 41, 475–511 (2006)
B. Buchberger, Groebner Bases:An Algorithmic Method in Polynomial Ideal Theory, in Multidimensional Systems Theory, ed. by N.K. Bose (D. Reidel Publishing, Dordrecht, 1985), pp. 184–232
M. Caboara, J. Perry, Reducing the size and number of linear programs in a dynamic Gröbner basis algorithm. Appl. Algebra Eng. Comm. Comput. 25, 99–117 (2014)
J. Canny, D. Manocha, Algorithm for implicitizing rational parametric surfaces. Comput. Aided Geom. Des. 9, 25–50 (1992)
S.-C. Chou, Mechanical Geometry Theorem Proving (D. Reidel Publishing, Dordrecht, 1988)
H. Clemens, A Scrapbook of Complex Curve Theory, 2nd edn. (American Mathematical Society, Providence, Rhode Island, 2002)
A. Cohen, H. Cuypers, H. Sterk (eds.), Some Tapas of Computer Algebra (Springer, Berlin-Heidelberg-New York, 1999)
S. Collart, M. Kalkbrener, D. Mall, Converting bases with the Gröbner walk. J. Symb. Comput. 24, 465–469 (1998)
D. Cox, J. Little, D. O’Shea, Using Algebraic Geometry, 2nd edn. (Springer, New York, 2005)
H.S.M. Coxeter, Regular Polytopes, 3rd edn. (Dover, New York, 1973)
J.H. Davenport, Y. Siret, E. Tournier, Computer Algebra, 2nd edn. (Academic, New York, 1993)
W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann, Singular 3-1-6—A computer algebra system for polynomial computations (2012), available at http://www.singular.uni-kl.de
W. Decker, G. Pfister, A First Course in Computational Algebraic Geometry. AIMS Library Series (Cambridge University Press, Cambridge, 2013)
H. Derksen, G. Kemper, Computational Invariant Theory (Springer, Berlin-Heidelberg-New York, 2002)
A. Dickenstein, I. Emiris (eds.), Solving Polynomial Equations (Springer, Berlin-Heidelberg-New York, 2005)
J. Draisma, E. Horobeţ, G. Ottaviani, B. Sturmfels, R. Thomas, The Euclidean distance degree of an algebraic variety (2013). arXiv:1309.0049 [math.AG]
M. Drton, B. Sturmfels, S. Sullivant, Lectures on Algebraic Statistics. Oberwohlfach Mathematical Seminars, vol. 39 (Birkhäuser, Basel-Boston-Berlin, 2009)
T.W. Dubé, The structure of polynomial ideals and Gröbner bases. SIAM J. Comput. 19, 750–775 (1990)
D. Dummit, R. Foote, Abstract Algebra, 3rd edn. (Wiley, New York, 2004)
C. Eder, J. Faugère, A survey on signature-based Gröbner basis algorithms (2014). arXiv:1404.1774 [math.AC]
D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, 3d corrected printing (Springer, New York-Berlin-Heidelberg, 1999)
D. Eisenbud, C. Huneke, W. Vasconcelos, Direct methods for primary decomposition. Invent. Math. 110, 207–235 (1992)
J. Farr, S. Gao, Gröbner bases and generalized Padé approximation. Math. Comp. 75, 461–473 (2006)
J. Faugère, A new efficient algorithm for computing Gröbner bases (F4). J. Pure Appl. Algebra 139, 61–88 (1999)
J. Faugère, Finding All the Solutions of Cyclic 9 Using Gröbner Basis Techniques, in Computer Mathematics (Matsuyama, 2001), Lecture Notes Ser. Comput., vol. 9 (World Scientific, River Edge, NJ, 2001), pp. 1–12
J. Faugère, A new efficient algorithm for computing Gröbner bases without reduction to zero F5. In: Proceedings of ISSAC’02, Villeneuve d’Ascq, France, July 2002, 15–82; revised version from http://www-polsys.lip6.fr/~jcf/Publications/index.html
J. Faugère, P. Gianni, D. Lazard, T. Mora, Efficient change of ordering for Gröbner bases of zero-dimensional ideals. J. Symb. Comput. 16, 329–344 (1993)
G. Fischer, Plane Algebraic Curves (AMS, Providence, Rhode Island, 2001)
J. Foley, A. van Dam, S. Feiner, J. Hughes,Computer Graphics: Principles and Practice, 2nd edn. (Addison-Wesley, Reading, MA, 1990)
W. Fulton, Algebraic Curves (W. A. Benjamin, New York, 1969)
M. Gallet, H. Rahkooy, Z. Zafeirakopoulos, On Computing the Elimination Ideal Using Resultants with Applications to Gröbner Bases (2013). arXiv:1307.5330 [math.AC]
J. von zur Gathen, J. Gerhard, Modern Computer Algebra, 3rd edn. (Cambridge University Press, Cambridge, 2013)
C.F. Gauss, Werke, vol. III (Königlichen Gesellschaft der Wissenschaften zu Göttingen, Göttingen, 1876)
R. Gebauer, H.M. Möller, On an Installation of Buchberger’s Algorithm, in Computational Aspects of Commutative Algebra, ed. by L. Robbiano (Academic Press, New York, 1988), pp. 141–152
I. Gelfand, M. Kapranov, A. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants (Birkhäuser, Boston, 1994)
P. Gianni, B. Trager, G. Zacharias, Gröbner bases and primary decomposition ofpolynomial ideals, in Computational Aspects of Commutative Algebra, ed. by L. Robbiano (Academic Press, New York, 1988), pp. 15–33
A. Giovini, T. Mora, G. Niesi, L. Robbiano, C. Traverso, “One sugar cube, please,” or Selection Strategies in the Buchberger Algorithm, in ISSAC 1991, Proceedings of the 1991International Symposium on Symbolic and Algebraic Computation, ed. by S. Watt (ACM Press, New York, 1991), pp. 49–54
M. Giusti, J. Heintz, La détermination des points isolés et de la dimension d’unevariété algébrique peut se faire en temps polynomial, in Computational Algebraic Geometryand Commutative Algebra, ed. by D. Eisenbud, L. Robbiano (Cambridge University Press, Cambridge, 1993), pp. 216–256
L. Glebsky, A proof of Hilbert’s Nullstellensatz Based on Groebner bases (2012). arXiv:1204.3128 [math.AC]
R. Goldman, Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces in Geometric Modeling (Morgan Kaufman, Amsterdam, Boston, 2003)
D. Grayson, M. Stillman, Macaulay2, a Software System for Research (2013), version 1.6, available at http://www.math.uiuc.edu/Macaulay2/
G.-M. Greuel, G. Pfister, A Singular Introduction to Commutative Algebra, 2nd edn. (Springer, New York, 2008)
P. Griffiths, Introduction to Algebraic Curves. Translations of Mathematical Monographs, vol. 76 (AMS, Providence, 1989)
P. Gritzmann, B. Sturmfels, Minkowski addition of polytopes: computational complexity and applications to Gröbner bases. SIAM J. Discrete Math. 6, 246–269 (1993)
J. Harris, Algebraic Geometry, A First Course, corrected edition (Springer, New York, 1995)
R. Hartshorne, Algebraic Geometry (Springer, New York, 1977)
G. Hermann, Die Frage der endlich vielen schritte in der theorie der polynomideale, Math. Ann. 95, 736–788 (1926)
A. Heyden, K. Åström, Algebraic properties of multilinear constraints. Math. Methods Appl. Sci. 20, 1135–1162 (1997)
D. Hilbert, Über die Theorie der algebraischen Formen, Math. Ann. 36, 473–534 (1890). Reprinted in Gesammelte Abhandlungen, vol. II (Chelsea, New York, 1965)
D. Hilbert, Theory of Algebraic Invariants (Cambridge University Press, Cambridge, 1993)
J. Hilmar, C. Smyth, Euclid meets Bézout: intersecting algebraic plane curves with the Euclidean algorithm. Am. Math. Monthly 117, 250–260 (2010)
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, II. Ann. Math. 79, 109–203, 205–326 (1964)
W.V.D. Hodge, D. Pedoe, Methods of Algebraic Geometry, vol. I and II (Cambridge University Press, Cambridge, 1968)
M. Jin, X. Li, D. Wang, A new algorithmic scheme for computing characteristic sets. J. Symb. Comput. 50, 431–449 (2013)
J. Jouanolou, Le formalisme du résultant. Adv. Math. 90, 117–263 (1991)
M. Kalkbrener, Implicitization by Using Gröbner Bases, Technical Report RISC-Series 90-27 (University of Linz, Austria, 1990)
K. Kendig, Elementary Algebraic Geometry, 2nd edn. (Dover, New York, 2015)
F. Kirwan, Complex Algebraic Curves. London Mathematical Society Student Texts, vol. 23 (Cambridge University Press, Cambridge, 1992)
F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vomFünften Grade (Teubner, Leipzig, 1884). English Translation, Lectures on the Ikosahedron and the Solution of Equations of the Fifth Degree (Trubner, London, 1888). Reprinted by Dover, New York (1956)
J. Koh, Ideals generated by quadrics exhibiting double exponential degrees. J. Algebra 200, 225–245 (1998)
M. Kreuzer, L. Robbiano, Computational Commutative Algebra, vol. 1 (Springer, New York, 2000)
M. Kreuzer, L. Robbiano, Computational Commutative Algebra, vol. 2 (Springer, New York, 2005)
T. Krick, A. Logar, An Algorithm for the Computation of the Radical of an Ideal in the Ring of Polynomials, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, ed. by H.F. Mattson, T. Mora, T.R.N. Rao. Lecture Notes in Computer Science, vol. 539 (Springer, Berlin, 1991), pp. 195–205
D. Lazard, Gröbner Bases, Gaussian Elimination and Resolution of Systems of Algebraic Equations, in Computer Algebra: EUROCAL 83, ed. by J.A. van Hulzen. Lecture Notes in Computer Science, vol. 162 (Springer, Berlin, 1983), pp. 146–156
D. Lazard, Systems of Algebraic Equations (Algorithms and Complexity), in Computational Algebraic Geometry and Commutative Algebra, ed. by D. Eisenbud, L. Robbiano (Cambridge University Press, Cambridge, 1993), pp. 84–105
M. Lejeune-Jalabert, Effectivité des calculs polynomiaux, Cours de DEA 1984–85, Institut Fourier, Université de Grenoble I (1985)
F. Macaulay, On some formulæin elimination. Proc. Lond. Math. Soc. 3, 3–27 (1902)
D. Manocha, Solving systems of polynomial equations. IEEE Comput. Graph. Appl. 14, 46–55 (1994)
Maple 18, Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario (2014). http://www.maplesoft.com
M. Marinari, H. Möller, T. Mora, Gröbner bases of ideals defined by functionals with an application to ideals of projective points. Appl. Algebra Eng. Comm. Comput. 4, 103–145 (1993)
Mathematica 10, Wolfram Research, Inc., Champaign, Illinois (2014). http://www.wolfram.com/mathematica
H. Matsumura, Commutative Ring Theory (Cambridge University Press, Cambridge, 1989)
E. Mayr, A. Meyer, The complexity of the word problem for commutative semigroups and polynomial ideals. Adv. Math. 46, 305–329 (1982)
R. Mines, F. Richman, W. Ruitenburg, A Course in Constructive Algebra (Springer, New York-Berlin-Heidelberg, 1988)
B. Mishra, Algorithmic Algebra. Texts and Monographs in Computer Science (Springer, New York-Berlin-Heidelberg, 1993)
H.M. Möller, F. Mora, Upper and Lower Bounds for the Degree of Groebner Bases, in EUROSAM 1984, ed. by J. Fitch. Lecture Notes in Computer Science, vol. 174 (Springer, New York-Berlin-Heidelberg, 1984), pp. 172–183
A. Montes, T. Recio, Generalizing the Steiner–Lehmus theorem using the Gröbner cover. Math. Comput. Simul. 104, 67–81 (2014)
A. Montes, M. Wibmer, Gröbner bases for polynomial systems with parameters. J. Symb. Comput. 45, 1391–1425 (2010)
D. Mumford, Algebraic Geometry I: Complex Projective Varieties, cCorrected 2nd printing (Springer, New York-Berlin-Heidelberg, 1981)
L. Pachter, B. Sturmfels (eds.), Algebraic Statistics for Computational Biology (Cambridge University Press, Cambridge, 2005)
R. Paul, Robot Manipulators: Mathematics, Programming and Control (MIT Press, Cambridge, MA, 1981)
G. Pistone, E. Riccomagno, H. Wynn, Algebraic Statistics: Computational Commutative Algebra in Statistics. Monographs on Statistics and Applied Probability, vol. 89 (Chapman and Hall, Boca Raton, FL, 2001)
L. Robbiano, On the theory of graded structures. J. Symb. Comp. 2, 139–170 (1986)
F. Rouillier, Solving zero-dimensional systems through the rational univariate representation. Appl. Algebra Eng. Comm. Comput. 5, 433–461 (1999)
P. Schauenburg, A Gröbner-based treatment of elimination theory for affine varieties. J. Symb. Comput. 42, 859–870 (2007)
A. Seidenberg, Constructions in algebra. Trans. Am. Math. Soc. 197, 273–313 (1974)
A. Seidenberg, On the Lasker–Noether decomposition theorem. Am. J. Math. 106, 611–638 (1984)
J.G. Semple, L. Roth, Introduction to Algebraic Geometry (Clarendon Press, Oxford, 1949)
I.R. Shafarevich, Basic Algebraic Geometry 1, 2, 3rd edn. (Springer, New York-Berlin-Heidelberg, 2013)
L. Smith, Polynomial Invariants of Finite Groups (A K Peters, Wellesley, MA, 1995)
W. Stein et al., Sage Mathematics Software, version 6.3. The Sage Development Team (2014), available at http://www.sagemath.org
B. Sturmfels, Computing final polynomials and final syzygies using Buchberger’s Gröbner bases method. Results Math. 15, 351–360 (1989)
B. Sturmfels, Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8 (American Mathematical Society, Providence, RI, 1996)
B. Sturmfels, Algorithms in Invariant Theory, 2nd edn. Texts and Monographs in Symbolic Computation (Springer, New York-Vienna, 2008)
I. Swanson, On the embedded primes of the Mayr-Meyer ideals. J. Algebra 275, 143–190 (2004)
C. Traverso, Hilbert functions and the Buchberger algorithm. J. Symb. Comput. 22, 355–376 (1997)
P. Ullrich, Closed-form formulas for projecting constructible sets in the theory of algebraically closed fields. ACM Commun. Comput. Algebra 40, 45–48 (2006)
B. van der Waerden, Moderne Algebra, Volume II (Springer, Berlin, 1931). English translations, Modern Algebra, Volume II (F. Ungar Publishing, New York, 1950); Algebra, Volume 2 (F. Ungar Publishing, New York, 1970); and Algebra, Volume II (Springer, New York-Berlin-Heidelberg, 1991). The chapter on Elimination Theory is included in the first three German editions and the 1950 English translation, but all later editions (German and English) omit this chapter
R. Walker, Algebraic Curves (Princeton University Press, Princeton, 1950). Reprinted by Dover, 1962
D. Wang, Elimination Methods, Texts and Monographs in Symbolic Computation (Springer, Vienna, 2001)
V. Weispfenning, Comprehensive Gröbner bases. J. Symb. Comput. 14, 1–29 (1992)
F. Winkler, On the complexity of the Gröbner bases algorithm overK[x, y, z], in EUROSAM 1984, ed. by J. Fitch. Lecture Notes in Computer Science, vol. 174 (Springer, New York-Berlin-Heidelberg, 1984), pp. 184–194
W.-T. Wu, On the decision problem and the mechanization of theorem-proving in elementary geometry, in Automated Theorem Proving: After 25 Years, ed. by W. Bledsoe, D. Loveland. Contemporary Mathematics, vol. 29 (American Mathematical Society, Providence, Rhode Island, 1983), pp. 213–234
W.-T. Wu, Mathematics Mechanization: Mechanical Geometry Theorem-Proving, Mechanical Geometry Problem-Solving and Polynomial Equations-Solving (Kluwer, Dordrecht, 2001)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Cox, D.A., Little, J., O’Shea, D. (2015). Elimination Theory. In: Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-16721-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-16721-3_3
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16720-6
Online ISBN: 978-3-319-16721-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)