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Numerical algebraic geometry and algebraic kinematics

Published online by Cambridge University Press:  28 April 2011

Charles W. Wampler
Affiliation:
General Motors Research and Development, Mail Code 480-106-359, 30500 Mound Road, Warren, MI 48090-9055, USA E-mail: Charles.W.Wampler@gm.com URL: www.nd.edu/˜cwample1
Andrew J. Sommese
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618, USA E-mail: sommese@nd.edu URL: www.nd.edu/˜sommese

Abstract

In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Allgower, E. L. and Georg, K. (1993), Continuation and path following. In Acta Numerica, Vol. 2, Cambridge University Press, pp. 164.Google Scholar
Allgower, E. L. and Georg, K. (1997), Numerical path following. In Handbook of Numerical Analysis, Vol. V, North-Holland, pp. 3207.Google Scholar
Allgower, E. L. and Georg, K. (2003), Introduction to Numerical Continuation Methods, Vol. 45 of Classics in Applied Mathematics, SIAM.Google Scholar
Alt, H. (1923), ‘Über die Erzeugung gegebener ebener Kurven mit Hilfe des Gelenkvierecks’, Z. Angew. Math. Mech. 3, 1319.Google Scholar
Angeles, J. (2007), Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms, third edition, Springer Science and Business Media.Google Scholar
Basu, S., Pollack, R. and Roy, M.-F. (2006), Algorithms in Real Algebraic Geometry, Vol. 10 of Algorithms and Computation in Mathematics, second edition, Springer.Google Scholar
Bates, D., Hauenstein, J. and Sommese, A. J. (2010 a), A parallel endgame. Preprint available at: www.nd.edu/~sommese/preprints.CrossRefGoogle Scholar
Bates, D., Hauenstein, J. and Sommese, A. J. (2011), Efficient pathtracking methods. To appear In Numerical Algorithms. Available at: www.nd.edu/~sommese.preprints.Google Scholar
Bates, D., Hauenstein, J., Peterson, C. and Sommese, A. J. (2009 a), ‘A numerical local dimension test for points on the solution set of a system of polynomial equations’, SIAM J. Numer. Anal. 47, 36083623.Google Scholar
Bates, D., Hauenstein, J., Peterson, C. and Sommese, A. J. (2010 b), Numerical decomposition of the rank-deficiency set of a matrix of multivariate polynomials. In Approximate Commutative Algebra, Vol. 14 of Texts and Monographs in Symbolic Computation, Springer, pp. 5577.Google Scholar
Bates, D., Hauenstein, J., Sommese, A. J. and Wampler, C. W. (2008 a), Software for numerical algebraic geometry: A paradigm and progress towards its implementation. In Software for Algebraic Geometry (Stillman, M., Takayama, N. and Verschelde, J., eds), Vol. 148 of IMA Volumes in Mathematics and its Applications, Springer, pp. 114.Google Scholar
Bates, D., Hauenstein, J., Sommese, A. J. and Wampler, C. W. (2008), Bertini: Software for numerical algebraic geometry. Available at: www.nd.edu/~sommese/bertini.Google Scholar
Bates, D., Hauenstein, J., Sommese, A. J. and Wampler, C. W. (2008 b), ‘Adaptive multiprecision path tracking’, SIAM J. Numer. Anal. 46, 722746.CrossRefGoogle Scholar
Bates, D., Hauenstein, J., Sommese, A. J. and Wampler, C. W. (2009 b), ‘Stepsize control for adaptive multiprecision path tracking’, Contemp. Math. 496, 21– 31.CrossRefGoogle Scholar
Bates, D., Peterson, C. and Sommese, A. J. (2006), ‘A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set’, J. Complexity 22, 475489.Google Scholar
Bates, D., Peterson, C. and Sommese, A. J. (2008 c), Applications of a numerical version of Terracini's lemma for secants and joins. In Algorithms in Algebraic Geometry (Dickenstein, A., Schreyer, F.-O. and Sommese, A. J., eds), Springer, pp. 114.Google Scholar
Beltrametti, M. C. and Sommese, A. J. (1995), The Adjunction Theory of Complex Projective Varieties, Vol. 16 of De Gruyter Expositions in Mathematics, De Gruyter.Google Scholar
Bennett, G. (1903), ‘A new mechanism’, Engineering 76, 777778.Google Scholar
Bonev, I. (2003), The true origins of parallel robots. Available at: www.parallemic.org/Reviews/Review007.html.Google Scholar
Bottema, O. and Roth, B. (1979), Theoretical Kinematics, Vol. 24 of North-Holland Series in Applied Mathematics and Mechanics, North-Holland. Reprinted by Dover (1990).Google Scholar
Boyse, J. and Gilchrist, J. (1982), ‘GMSOLID: Interactive modeling for design and analysis of solids’, IEEE Comput. Graphics Appl. 2, 2740.Google Scholar
Carricato, M. and Parenti-Castelli, V. (2003), ‘A family of 3-DOF translational parallel manipulators’, J. Mech. Design 125, 302307.Google Scholar
Cayley, A. (1876), ‘On three-bar motion’, Proc. London Math. Soc. VII, 136166.Google Scholar
Chebyshev, P. (1854), ‘Théorie des mécanismes connus sous le nom de par-allélogrammes’, Mémoires des Savants Étrangers Présentés à l'Académie de Saint-Pétersbourg 7, 539568.Google Scholar
Chow, S. N., Mallet-Paret, J. and Yorke, J. A. (1979), A homotopy method for locating all zeros of a system of polynomials. In Functional Differential Equations and Approximation of Fixed Points (Bonn 1978), Vol. 730 of Lecture Notes in Mathematics, Springer, pp. 7788.Google Scholar
Clifford, W. (1878), ‘On the triple generation of three-bar curves’, Proc. London Math. Soc. 9, 2728.Google Scholar
Cox, D., Little, J. and O'Shea, D. (1997), Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, second edition, Springer.Google Scholar
Cox, D., Little, J. and O'Shea, D. (1998), Using Algebraic Geometry, Vol. 185 of Graduate Texts in Mathematics, Springer.Google Scholar
Dayton, B. and Zeng, Z. (2005), Computing the multiplicity structure in solving polynomial systems. In Proc. ISSAC 2005, ACM, pp. 166–123.Google Scholar
Dayton, B. H., Li, T.-Y. and Zeng, Z. (2011), Multiple zeros of nonlinear systems. To appear In Math. Comp. 80. Available at: www.ams.org/journals/mcom/2011–80–275/.CrossRefGoogle Scholar
de Groot, J. (1970), Bibliography on Kinematics, Eindhoven University of Technology.Google Scholar
Delassus, E. (1922), ‘Les chaânes articulées fermées at déformables à quatre membres’, Bull. Sci. Math. Astronom. 46, 283304.Google Scholar
Di Gregorio, R. and Parenti-Castelli, V. (1998), A translational 3-DOF parallel manipulator. In Advances in Robot Kinematics: Analysis and Control (Lenarcic, J. and Husty, M. L., eds), Kluwer Academic, pp. 4958.Google Scholar
Dijksman, E. (1976), Motion Geometry of Mechanisms, Cambridge University Press.Google Scholar
Drexler, F. J. (1977), ‘Eine Methode zur Berechnung sämtlicher Lösungen von Polynomgleichungssystemen’, Numer. Math. 29, 4558.Google Scholar
Duffy, J. and Crane, C. (1980), ‘A displacement analysis of the general spatial 7-link, 7R mechanism’, Mech. Mach. Theory 15, 153169.Google Scholar
Freudenstein, F. and Roth, B. (1963), ‘Numerical solution of systems of nonlinear equations.’, J. Assoc. Comput. Mach. 10, 550556.Google Scholar
Freudenstein, F. and Sandor, G. (1959), ‘Synthesis of path-generating mechanisms by means of a programmed digital computer’, ASME J. Engng Ind. 81, 159168.Google Scholar
Garcia, C. B. and Li, T. Y. (1980), ‘On the number of solutions to polynomial systems of equations’, SIAM J. Numer. Anal. 17, 540546.Google Scholar
Garcia, C. B. and Zangwill, W. I. (1979), ‘Finding all solutions to polynomial systems and other systems of equations’, Math. Programming 16, 159176.Google Scholar
Geiss, F. and Schreyer, F.-O. (2009), A family of exceptional Stewart–Gough mechanisms of genus 7. In Interactions of Classical and Numerical Algebraic Geometry (Bates, D., Besana, G.-M., Rocco, S. D. and Wampler, C. W., eds), Vol. 496 of Contemporary Mathematics, AMS, pp. 221234.Google Scholar
Gogu, G. (2004), ‘Structural synthesis of fully-isotropic translational parallel robots via theory of linear transformations’, Europ. J. Mechanics A/Solids 23, 10211039.Google Scholar
Griffiths, P. and Harris, J. (1994), Principles of Algebraic Geometry, Wiley Classics Library, Wiley.Google Scholar
Halsted, G. (1895), ‘Biography: Pafnutij Lvovitsch Tchebychev’, Amer. Math. Monthly 2.Google Scholar
Hartshorne, R. (1977), Algebraic Geometry, Vol. 52 of Graduate Texts in Mathematics, Springer.Google Scholar
Hauenstein, J., Sommese, A. J. and Wampler, C. W. (2010), Regenerative cascade homotopies for solving polynomial systems. Preprint available at: www.nd.edu/~sommese/preprints.CrossRefGoogle Scholar
Hauenstein, J., Sommese, A. J. and Wampler, C. W. (2011), ‘Regeneration homo-topies for solving systems of polynomials’, Math. Comp. 80, 345377.Google Scholar
Huang, T., Li, Z., Li, M., Chetwynd, D. and Gosselin, C. (2004), ‘Conceptual design and dimensional synthesis of a novel 2-DOF translational parallel robot for pick-and-place operations’, J. Mech. Design 126, 449455.Google Scholar
Hunt, K. (1978), Kinematic Geometry of Mechanisms, Clarendon Press.Google Scholar
Husty, M. L. (1996), ‘An algorithm for solving the direct kinematics of general Stewart–Gough platforms’, Mech. Mach. Theory 31, 365380.Google Scholar
Husty, M. L. and Karger, A. (2000), Self-motions of Griffis–Duffy type parallel manipulators. In Proc. 2000 IEEE Int. Conference on Robotics and Automation (San Francisco 2000), pp. 712.Google Scholar
Husty, M. L., Karger, A., Sachs, H. and Steinhilper, W. (1997), Kinematik und Robotik, Springer.Google Scholar
Husty, M. L., Pfurner, M., Schröcker, H.-P. and Brunnthaler, K. (2007), ‘Algebraic methods in mechanism analysis and synthesis’, Robotica 25, 661675.Google Scholar
Karger, A. (2003), ‘Architecture singular planar parallel manipulators’, Mech. Mach. Theory 38, 11491164.Google Scholar
Karger, A. (2008), ‘Architecturally singular non-planar parallel manipulators’, Mech. Mach. Theory 43, 335346.CrossRefGoogle Scholar
Kempe, A. (1877), How to Draw a Straight Line: A Lecture on Linkages, Macmillan. Available at: www2.cddc.vt.edu/gutenberg/2/5/1/5/25155/25155-pdf.pdf.Google Scholar
Kong, X. and Gosselin, C. (2002), ‘Kinematics and singularity analysis of a novel type of 3-CRR 3-DOF translational parallel manipulator’, Int. J. Robotics Research 21, 791798.Google Scholar
Kong, X. and Gosselin, C. (2004), ‘Type synthesis of 3-DOF translational parallel manipulators based on screw theory’, J. Mech. Design 126, 8392.Google Scholar
Lazard, D. (1993), On the representation of rigid-body motions and its application to generalized platform manipulators. In Computational Kinematics (Angeles, J., Kovacs, P. and Hommel, G., eds), Kluwer, pp. 175182.Google Scholar
Lee, T.-L., Li, T.-Y. and Tsai, C. (2008), ‘HOM4PS-2.0: A software package for solving polynomial systems by the polyhedral homotopy continuation method’, Computing 83, 109133.Google Scholar
Leykin, A., Verschelde, J. and Zhao, A. (2006), ‘Newton's method with deflation for isolated singularities of polynomial systems’, Theor. Comp. Sci. 359, 111– 122.Google Scholar
Leykin, A., Verschelde, J. and Zhao, A. (2008), Higher-order deflation for polynomial systems with isolated singular solutions. In Algorithms in Algebraic Geometry (Dickenstein, A., Schreyer, F.-O. and Sommese, A. J., eds), Springer, pp. 7997.Google Scholar
Li, T.-Y. (1997), Numerical solution of multivariate polynomial systems by homo-topy continuation methods. In Acta Numerica, Vol. 6, Cambridge University Press, pp. 399436.Google Scholar
Li, T.-Y. (2003), Numerical solution of polynomial systems by homotopy continuation methods. In Handbook of Numerical Analysis, Vol. XI, North-Holland, pp. 209304.Google Scholar
Li, T.-Y., Sauer, T. and Yorke, J. A. (1989), ‘The cheater's homotopy: An efficient procedure for solving systems of polynomial equations’, SIAM J. Numer. Anal. 26, 12411251.CrossRefGoogle Scholar
Li, Y. and Xu, Q. (2006), ‘Kinematic analysis and design of a new 3-DOF translational parallel manipulator’, J. Mech. Design 128, 729737.Google Scholar
Lu, Y., Bates, D., Sommese, A. J. and Wampler, C. W. (2007), Finding all real points of a complex curve. In Proc. Midwest Algebra, Geometry and its Interactions Conference, Vol. 448 of Contemporary Mathematics, AMS, pp. 183205.Google Scholar
McCarthy, J. (2000), Geometric Design of Linkages, Springer.Google Scholar
Morgan, A. P. (1986), ‘A transformation to avoid solutions at infinity for polynomial systems’, Appl. Math. Comput. 18, 7786.Google Scholar
Morgan, A. P. (1987), Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems, Prentice Hall.Google Scholar
Morgan, A. P. and Sommese, A. J. (1987 a), ‘A homotopy for solving general polynomial systems that respects m-homogeneous structures’, Appl. Math. Comput. 24, 101113.Google Scholar
Morgan, A. P. and Sommese, A. J. (1987 b), ‘Computing all solutions to polynomial systems using homotopy continuation’, Appl. Math. Comput. 24, 115138. Errata: Appl. Math. Comput. 51 (1992), 209.Google Scholar
Morgan, A. P. and Sommese, A. J. (1989), ‘Coefficient-parameter polynomial continuation’, Appl. Math. Comput. 29, 123160. Errata: Appl. Math. Comput. 51 (1992), 207.Google Scholar
Morgan, A. P., Sommese, A. J. and Wampler, C. W. (1991), ‘Computing singular solutions to nonlinear analytic systems’, Numer. Math. 58, 669684.Google Scholar
Morgan, A. P., Sommese, A. J. and Wampler, C. W. (1992 a), ‘Computing singular solutions to polynomial systems’, Adv. Appl. Math. 13, 305327.Google Scholar
Morgan, A. P., Sommese, A. J. and Wampler, C. W. (1992 b), ‘A power series method for computing singular solutions to nonlinear analytic systems’, Numer. Math. 63, 391409.CrossRefGoogle Scholar
Morgan, A. P., Sommese, A. J. and Wampler, C. W. (1995), ‘A product-decomposition bound for Bézout numbers’, SIAM J. Numer. Anal. 32, 13081325.Google Scholar
Mourrain, B. (1993), The 40 generic positions of a parallel robot. In Proc. ISSAC′93 (Bronstein, M., ed.), ACM Press, pp. 173182.Google Scholar
Ojika, T. (1987), ‘Modified deflation algorithm for the solution of singular problems I: A system of nonlinear algebraic equations’, J. Math. Anal. Appl. 123, 199221.CrossRefGoogle Scholar
Ojika, T., Watanabe, S. and Mitsui, T. (1983), ‘Deflation algorithm for the multiple roots of a system of nonlinear equations’, J. Math. Anal. Appl. 96, 463479.CrossRefGoogle Scholar
Raghavan, M. (1991), The Stewart platform of general geometry has 40 configurations. In Proc. ASME Design and Automation Conference, Vol. 32–2, ASME, pp. 397402.Google Scholar
Raghavan, M. (1993), ‘The Stewart platform of general geometry has 40 configurations’, ASME J. Mech. Design 115, 277282.Google Scholar
Roberts, S. (1875), ‘On three-bar motion in plane space’, Proc. London Math. Soc. VII, 1423.Google Scholar
Ronga, F. and Vust, T. (1995), Stewart platforms without computer? In Real Analytic and Algebraic Geometry (Trento 1992), De Gruyter, pp. 197212.Google Scholar
Roth, B. and Freudenstein, F. (1963), ‘Synthesis of path-generating mechanisms by numerical means’, J. Engng Ind., Trans. ASME, 85, 298306.Google Scholar
Schönflies, A. and Greubler, M. (1902), Kinematik. In Enzyclopaedie der Mathematischen Wissenschaften, Vol. 3, Teubner, pp. 190278.Google Scholar
Selig, J. (2005), Geometric Fundamentals of Robotics, second edition, Monographs in Computer Science, Springer.Google Scholar
Sommese, A. J. and Verschelde, J. (2000), ‘Numerical homotopies to compute generic points on positive dimensional algebraic sets’, J. Complexity 16, 572602.Google Scholar
Sommese, A. J. and Wampler, C. W. (1996), Numerical algebraic geometry. In The Mathematics of Numerical Analysis (Park City, UT, 1995), Vol. 32 of Lectures in Applied Mathematics, AMS, pp. 749763.Google Scholar
Sommese, A. J. and Wampler, C. W. (2005), The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, World Scientific.Google Scholar
Sommese, A. J. and Wampler, C. W. (2008), ‘Exceptional sets and fiber products’, Foundations of Computational Mathematics 28, 171196.Google Scholar
Sommese, A. J., Verschelde, J. and Wampler, C. W. (2001 a), ‘Numerical decomposition of the solution sets of polynomial systems into irreducible components’, SIAM J. Numer. Anal. 38, 20222046.Google Scholar
Sommese, A. J., Verschelde, J. and Wampler, C. W. (2001 b), Numerical irreducible decomposition using projections from points on the components. In Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering (South Hadley, MA, 2000), Vol. 286 of Contemporary Mathematics, AMS, pp. 3751.CrossRefGoogle Scholar
Sommese, A. J., Verschelde, J. and Wampler, C. W. (2001 c), Using monodromy to decompose solution sets of polynomial systems into irreducible components. In Applications of Algebraic Geometry to Coding Theory, Physics and Computation (Eilat 2001), Vol. 36 of NATO Sci. Ser. II Math. Phys. Chem., Kluwer, pp. 297315.Google Scholar
Sommese, A. J., Verschelde, J. and Wampler, C. W. (2002 a), A method for tracking singular paths with application to the numerical irreducible decomposition. In Algebraic Geometry, De Gruyter, pp. 329345.Google Scholar
Sommese, A. J., Verschelde, J. and Wampler, C. W. (2002 b), ‘Symmetric functions applied to decomposing solution sets of polynomial systems’, SIAM J. Numer. Anal. 40, 20262046.Google Scholar
Sommese, A. J., Verschelde, J. and Wampler, C. W. (2004 a), ‘Advances in polynomial continuation for solving problems in kinematics’, J. Mech. Design 126, 262– 268.CrossRefGoogle Scholar
Sommese, A. J., Verschelde, J. and Wampler, C. W. (2004 b), ‘Homotopies for intersecting solution components of polynomial systems’, SIAM J. Numer. Anal. 42, 15521571.Google Scholar
Sommese, A. J., Verschelde, J. and Wampler, C. W. (2005), ‘An intrinsic homotopy for intersecting algebraic varieties’, J. Complexity 21, 593608.Google Scholar
Sommese, A. J., Verschelde, J. and Wampler, C. W. (2008), Solving polynomial systems equation by equation. In Algorithms in Algebraic Geometry, Vol. 146 of IMA Volumes in Mathematics and its Applications, Springer, pp. 133152.Google Scholar
Stetter, H. J. (2004), Numerical Polynomial Algebra, SIAM.Google Scholar
Study, E. (1891), ‘Von den Bewegungen und Umlegungen’, Mathematische Annalen 39, 441556.Google Scholar
Study, E. (1903), Geometrie der Dynamen, Teubner.Google Scholar
Su, H.-J., McCarthy, J. and Watson, L. (2004), ‘Generalized linear product homotopy algorithms and the computation of reachable surfaces’, J. Comput. Inf. Sci. Engng 4, 226234.CrossRefGoogle Scholar
Su, H.-J., McCarthy, J., Sosonkina, M. and Watson, L. (2006), ‘Algorithm 857. POL-SYS GLP: A parallel general linear product homotopy code for solving polynomial systems of equations’, ACM Trans. Math. Software 32, 561579.CrossRefGoogle Scholar
Sylvester, J. (1874), ‘On recent discoveries in mechanical conversion of motion’, Proc. Royal Institution of Great Britain 7, 179198.Google Scholar
Tari, H., Su, H.-J. and Li, T.-Y. (2010), ‘A constrained homotopy technique for excluding unwanted solutions from polynomial equations arising in kinematics problems’, Mech. Mach. Theory 45, 898910.CrossRefGoogle Scholar
Tsai, L.-W. and Morgan, A. P. (1985), ‘Solving the kinematics of the most general six- and five-degree-of-freedom manipulators by continuation methods’, ASME J. Mech., Trans., Auto. Design 107, 4857.Google Scholar
Tsai, L.-W., Walsh, G. and Stamper, R. (1996), Kinematics of a novel three DOF translational platform. In Proc. 1996 IEEE Int. Conf. Robotics and Automation, Vol. 4, pp. 3446 –3451.Google Scholar
Verschelde, J. (1999), ‘Algorithm 795. PHCpack: A general-purpose solver for polynomial systems by homotopy continuation’, ACM Trans. Math. Software 25, 251276.Google Scholar
Verschelde, J. and Cools, R. (1993), ‘Symbolic homotopy construction’, Appl. Algebra Engng Comm. Comput. 4, 169183.Google Scholar
Wampler, C. W. (1996), ‘Forward displacement analysis of general six-in-parallel SPS (Stewart) platform manipulators using soma coordinates’, Mech. Mach. Theory 31, 331337.Google Scholar
Wampler, C. W., Hauenstein, J. and Sommese, A. J. (2011), Mechanism mobility and a local dimension test. To appear In Mech. Mach. Theory. Available at: www.nd.edu/~sommese/preprints.Google Scholar
Wampler, C. W., Morgan, A. P. and Sommese, A. J. (1992), ‘Complete solution of the nine-point path synthesis problem for four-bar linkages’, ASME J. Mech. Design 114, 153159.CrossRefGoogle Scholar
Wampler, C. W., Morgan, A. P. and Sommese, A. J. (1997), ‘Complete solution of the nine-point path synthesis problem for four-bar linkages: Closure’, ASME J. Mech. Design 119, 150152.Google Scholar
Watson, L. T., Sosonkina, M., Melville, R. C., Morgan, A. P. and Walker, H. F. (1997), ‘Algorithm 777. HOMPACK90: A suite of Fortran 90 codes for globally convergent homotopy algorithms’, ACM Trans. Math. Software 23, 514549.Google Scholar
Wise, S. M., Sommese, A. J. and Watson, L. T. (2000), ‘Algorithm 801. POL-SYS PLP: A partitioned linear product homotopy code for solving polynomial systems of equations’, ACM Trans. Math. Software 26, 176200.CrossRefGoogle Scholar
Zeng, Z. (2009), The closedness subspace method for computing the multiplicity structure of a polynomial system. In Interactions of Classical and Numerical Algebraic Geometry, Vol. 496 of Contemporary Mathematics, AMS, pp. 347362.Google Scholar