Solving Polynomial Equation Systems III

Teo Mora

doi:10.1017/CBO9781139015998

References

Bibliography

Aho, A.V., Hopcroft, J.E., Ullman, J.D., The Design and Analysis of Computer Algorithms, Addison–Wesley (1974 Google Scholar).

Alonso, M.E., Becker, E., Roy, M.-F., Wörmann T., Zeroes, multiplicities and idempotents for zero dimensional systems, in Prog. Math. 143 (1996 Google Scholar), pp. 1–16, Birkhäuser.

Ampère, A.-M., Fonctions Interpolaires, Annales de M. Gergonne (1826 Google Scholar).

Arnaudiès, J.M., Valibouze, A., Résolventes de Lagrange, Report LIPT 93.61 (1993 Google Scholar),

Arnaudiès, J.M., Valibouze, A., Lagrange resolvents, J. Pure Appl. Algebra 117–118 (1996 Google Scholar), 23–40.

Aubry, P., Moreno Maza, M., Triangular set for solving polynomial systems: a comparative implementation of four methods, J. Symb. Comp. 28 (1999 Google Scholar), 125–154.

Aubry, P., Valibouze, A., Using Galois ideals for computing relative resolvents, J. Symb. Comp. 30 (2000 Google Scholar), 635–651.

Aubry, P., Lazard, D., Moreno Maza, M., On the theories of triangular sets, J. Symb. Comp. 28 (1999 Google Scholar), 105–124.

Auzinger, W., Stetter, H.J., An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations, in I.S.N.M. 86 (1988 Google Scholar), pp. 11–30, Birkhäuser.

Becker, E., Cardinal, J.-P., Roy, M.-F., Szafraniec, Z., Multivariate Bezoutians, Kronecker symbol and Eisenbud–Levin formula, in Prog. Math. 143 (1996 Google Scholar), pp. 79–104, Birkhäuser.

Bézout, E., Recherches sur le degré des équations résultantes de l'évanouissement des inconnues, et sur les moyens qu'il convient d'employer pour trouver ses équations, Mém. Acad. Roy. Sci. Paris (1764 Google Scholar), 288–233.

Bézout, E., Théorie Generale des Èquations Algébriques, Pierres, Paris (1771 Google Scholar).

Bini, D., Pan, V., Polynomial and Matrix Computations, Birkhäuser (1994 Google Scholar).

Bostajn, A., Salvy, B., Schost, E., Fast algorithm for zero-dimensional polynomial systems using duality, J. AAECC 14 (2003 Google Scholar), 239–272.

Bürgisser, P., Clausen, M., Shorolahi, M.A., Algebraic Complexity Theory, Springer (1997 Google Scholar).

Burnside, W., Theory of Groups of Finite Order, Cambridge University Press (1911 Google Scholar).

Canny, J., Generalized characteristic polynomials, in L. N. Comp. Sci. 358 (1988 Google Scholar), pp. 293–299, Springer.

Canny, J., An effective algorithm for the sparse mixed resultant, in L. N. Comp. Sci. 673 (1993 Google Scholar), pp. 89–104, Springer.

Cardinal, J.P., Dualité et algorithms itératifs pour la résolution de systémes polynomiaux, Ph.D.thesis, University of Rennes I (1993 Google Scholar).

Cardinal, J.P., Mourrain, B., Algebraic approach of residues and applications, in L. N. Appl. Math. 32 (1999 Google Scholar), American Mathematical Society Press.

Cauchy, A., Usage des fonctions interpolaires dans ls determination des fonctions symmetriques des racines d'une équation algébrique donnée, C.R. Acad. Sci. Paris 11 (1840 Google Scholar), 933.

Cauchy, A., Oeuvres t. V, Gauthier–Villars, Paris, (1882 Google Scholar).

Cayley, A., On the theory of elimination, Camb. Dublin Math. J. III (1848 Google Scholar), 116–120.

Cayley, A., Note sur la méthode d'élimination de Bezout, J. Reine Ang. Math. LIII (1857 Google Scholar), 366–367.

Cayley, A., A fourth memoir upon quantics, Phil. Trans. Royal Soc. London CXLVIII (1858 Google Scholar), 415–427.

Charden, M., Un algorithm pour les calcul des resultants, in Prog. Math. 94 (1990 Google Scholar), pp. 47–62, Birkhäuser.

Charden, M., The resultant via a Koszul complex, Prog. Math. 109 (1993 Google Scholar), pp. 29–40, Birkhäuser.

Chen, C., Golubitsky, O., Lemaire, F., Moreno Maza, M., Comprehensive triangular decomposition, in Proc. CASC 2007 (2007 Google Scholar), pp. 73–101.

Dahan, X., Sur la complexité des représentations des systèmes polynomiaux: triangulation, méthodes modulaires, évaluation dynamique, Ph.D. thesis, École Polytechnique (2006 Google Scholar).

Dahan, X., Schost, E., Sharp estimates for triangular sets, in Proc. ISSAC'04 (2004 Google Scholar), pp. 103–110, Association for Computing Machinery.

Dahan, X., Moreno Maza, M., Schost, E., Wu, W., Xie, Y., Lifting techniques for triangular decomposition, in Proc. ISSAC'05 (2005 Google Scholar), pp. 108–115.

Delassus, E., Sur les systèmes algébriques et leurs relations avec certains systèmes d'equations aux dérivées partielles. Ann. Éc. Norm. 3e série 14 (1897 Google Scholar), 21–44.

Dixon, A.L., On a form of the eliminant of two quantics, Proc. London Math. Soc. 6 (1908 Google Scholara), 468–478.

Dixon, A.L., The eliminant of three quantics in two independent variables, Proc. London Math. Soc. 7 (1908 Google Scholarb), 49–69.

Dixon, A.L., Some results in the theory of elimination, Proc. Roy. Soc. London 82 (1909 Google Scholar), 468–478.

Euler, L., Introductio in Analysin Infinitorum, Tom. 2, Lausanne, (1748 Google Scholar).

Felszeghy, B., Ráth, B., Rónyai, L., The lex game and some applications, J. Symb. Comp. 41 (2006 Google Scholar), 663–681.

Gallo, G., Mishra, B., Effective algorithms and bounds for Wu–Ritt characteristic sets, in Prog. Math. 94 (1990 Google Scholar), pp. 119–142, Birkhäuser.

Gallo, G., Mishra, B., A solution to Kronecker's problem, J. AAECC 5 (1994 Google Scholar), 343–370.

Gallo, G., Mishra, B., Olivier, F., Some constructions in rings of differential polynomials, in L. N. Comp. Sci. 539 (1991 Google Scholar), pp. 171–182, Springer.

Gianni, P., Properties of GrÖbner bases under specialization, in L. N. Comp. Sci. 378 (1987 Google Scholar), pp. 293–297, Springer.

Giusti, M., Heintz, J., Algorithmes – disons rapides – pour la décomposition d'une variété algébrique en composantes irréductibles, in Prog. Math. 94 (1990 Google Scholar), pp. 169–194, Birkhäuser.

Giusti, M., Heintz, J., La detérmination des point isolés et de la dimension d'une variété algébrique peut se faire en temps polynomial, in Symp. Math. 34 (1993 Google Scholar), pp. 216–256, Cambridge University Press.

Giusti, M., Schost, E., Solving some overdetermined polynomial systems, in Proc. ISSAC'99 (1999 Google Scholar), pp. 1–8, Association for Computing Machinery.

Giusti, M., Heintz, J., Morais, J.E., Pardo, L.M., When polynomial equation systems can be “solved” fast?, in L. N. Comp. Sci. 948 (1995 Google Scholar), pp. 205–231, Springer.

Giusti, M., Heintz, J., Hägele, K., Morais, J.E., Pardo, L.M., Montaña, J.M., Lower bounds for diophantine approximation, J. Pure Appl. Algebra 117–118 (1997 Google Scholara), 277–311.

Giusti, M., Heintz, J., Morais, J.E., Pardo, L.M., Le rôle des structures de données dans les problèmes d'élimination, C.R. Acad. Sci. Paris 325 (1997 Google Scholarb), 1223–1228.

Giusti, M., Heintz, J., Morais, J.E., Morgensten, J., Pardo, L.M., Straight-line programs in geometric elimination theory, J. Pure Appl. Algebra 124 (1998 Google Scholar), 101–146.

Giusti, M., Hägele, K., Lecerf, G., Marchand, J., Salvy, B., The projective Noether maple package: computing the dimension of a projective variety, J. Symb. Comp. 30 (2000 Google Scholar), 291–307.

Giusti, M., Lecerf, G., Salvy, B., A Gröbner free alternative for polynomial system solving, J. Complexity 17 (2001 Google Scholar), 154–211.

Gonzalez-Vega, L., Rouiller, F., Roy, M.-F., Symbolic recipes for polynomial system solving, in Some Tapas of Computer Algebra, ed. A., Cohen, (1997 Google Scholar), Springer.

Gunther, N., Sur la forme canonique des systèmes équations homogènes (in Russian)(Journal de l'Institut des Ponts et Chaussées de Russie), Izdanie Inst. In?z. Putej Soob?s?cenija Imp. Al. I. 84 (1913 Google Scholar).

Gunther, N., Sur les caractéristiques des systémes d'équations aux dérivées partialles, C.R. Acad. Sci. Paris 156 (1913 Google Scholar), 1147–1150.

Gunther, N., Sur la forme canonique des equations algébriques, C.R. Acad. Sci. Paris 157 (1913 Google Scholar), 577–80.

Hägele, K., Morais, J.E., Pardo, L.M., Sombra, M., On the intrinsic complexity of the arithmetic Nullstellensatz, J. Pure Appl. Algebra 146 (2000 Google Scholar), 103–183.

Hashemi, A., Structure et Complexité des bases de Gröbner, J. Symb. Comp. 45 (2010 Google Scholar), 1330–1340.

Hashemi, A., Lazard, D., Sharper complexity bounds for zero-dimensional Gröbner bases and polynomial system solving, Int. J. Algebra Comp. 21 (2011 Google Scholar), 705–713.

Jacobi, C.G.I., De eliminatione variabilis e duabus aequationibus algebraicas, J. Reine Ang. Math. XV (1836 Google Scholar) 101–124.

Jouanoulou, J.-P., Le formalisme du résultant, Adv. Math. 90 Google Scholar 117–263.

Kalkbrener, M., Solving systems of algebraic equations by using Gröbner bases, in L. N. Comp. Sci. 378 (1987 Google Scholar), pp. 282–292, Springer.

Kalkbrener, M., Three contributions to elimination theory, Ph.D. thesis, Linz University (1991 Google Scholar).

Kalkbrener, M., A generic euclidean algorithm for computing triangular representations of algebraic varieties, J. Symb. Comp. 15 (1993 Google Scholar), 153–167.

Kalkbrener, M., On the stability of Gröbner bases under specialization, J. Symb. Comp. 24 (1997 Google Scholar), 51–58.

Kapur, D., Cai, Y., An algorithm for computing a Gröbner basis of a polynomial ideal over a ring with zero divisors, Math. Comput. Sci. 2 (2009 Google Scholar), 601–634.

Kapur, D., Chtcherba, A.D., Conditions for exact resultants using the Dixon resultant formulation, in Proc. ISSAC 2000 (2000 Google Scholar), pp. 62–70, Association for Computing Machinery.

Kapur, D., Chtcherba, A.D., On the efficiency and optimality of Dixon-based resultant method, in Proc. ISSAC 2002 (2002 Google Scholar), pp. 29–36, Association for Computing Machinery.

Kapur, D., Saxena, T., Yang, L., Algebraic and geometric reasoning using Dixon resultants, in Proc. ISSAC 94 (1994 Google Scholar), pp. 99–136, Association for Computing Machinery.

Kapur, D., Saxena, T., Extraneous factors in the Dixon resultant formulation, in Proc. ISSAC 97 (1997 Google Scholar), pp. 141–148, Association for Computing Machinery.

Kobayashi, H., Moritsugu, S., Hogan, R.W., On radical zero-dimensional ideals, J. Symb. Comp. 8 (1989 Google Scholar), 545–552.

Kratzer, M., Computing the dimension of a polynomial ideal and membership in lowdimensional ideals. Master's thesis, Technische Universität München (2008 Google Scholar).

Krick, T., Pardo, L.M., Une approache informatique pour l'approximation diophantienne, C.R. Acad. Sci. Paris 318 (1994 Google Scholar), 407–412.

Krick, T., Pardo, L.M., A computational method for Diphantine approximation, in Prog. Math. 143 (1996 Google Scholar), pp. 193–254, Birkhäuser.

Lakshman|Y.N., Lazard, D., On the complexity of zero-dimensional algebraic systems, in Prog. Math. 94 (1990 Google Scholar), pp. 217–226, Birkhäuser.

Lazard, D., Algèbre linéaire sur K[X1, …, Xn] et élimination, Bull. Soc. Math. France 105 (1977 Google Scholar), 165–190.

Lazard, D., Systems of algebraic equations, in L. N. Comp. Sci. 72 (1979 Google Scholar), pp. 88–94, Springer.

Lazard, D., Resolution des systemes d'equations algebriques, Theoret. Comput. Sci. 15 (1981 Google Scholar), 77–110.

Lazard, D., A new method for solving algebraic systems of positive dimension, Disc. Appl. Math. 33 (1991 Google Scholar), 147–160.

Lazard, D., Solving zero-dimensional algebraic systems, J. Symb. Comp. 15 (1992 Google Scholar), 117–132.

Lazard, D., Systems of algebraic equations (algorithms and complexity), Symp. Math. 34 (1993 Google Scholar), 84–106, Cambridge University Press.

Lazard, D., Resolution of polynomial systems, in Proc. ASCM 2000 (2000 Google Scholar), pp. 1–8, World Scientific.

Lazard, D., On the specification for solvers of polynomial systems, in Proc. ASCM 2001 (2001 Google Scholar), pp. 1–10, World Scientific.

Lazard, D., Thirty years of polynomial system solving, and now?, J. Symb. Comp. 44 (2009 Google Scholar), 222–239.

Lazard, D., Rouillier, F., Solving parametric polynomial systems, J. Symb. Comp. 42 (2007 Google Scholar), 636–667.

Lecerf, G., Une alternative aux méthodes de réécriture pour résolution des systémes algébriques, Ph.D. thesis, École Polytechnique (2001 Google Scholar).

Lundqvist, S., Complexity of comparing monomials and two improvements of the BMalgorithm, in L. N. Comp. Sci. 5393 (2008 Google Scholar), pp. 105–125, Springer.

Lundqvist, S., Vector space bases associated to vanishing ideals of points, J. Pure Appl. Algebra 214 (2010 Google Scholar), 309–321.

Macaulay, F. S., Some formulae in elimination, Proc. London Math. Soc. (1) 35 (1903 Google Scholar), 3–27.

Macaulay, F. S., The Algebraic Theory of Modular Systems, Cambridge University Press (1916 Google Scholar).

Malle, G., Trinks, W., Zur Behandlung algebraischer Gleichungssysteme mit dem Computer, preprint (1985 Google Scholar).

Manocha, D., Cannon, J.F., Multipolynomial resultant algorithms, J. Symb. Comp. 15 (1993 Google Scholar), 99–122.

Möller, H.M., Systems of algebraic equations solved by means of endomorphisms, in L. N. Comp. Sci. 673 (1993 Google Scholara), pp. 43–56, Springer.

Möller, H.M., On decomposing systems of polynomial equations with finitely many solutions, J. AAECC 4 (1993 Google Scholarb), 217–230.

Möller, M., Stetter, H., Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems, Num. Math. 70 (1995 Google Scholar), 311–325.

Monico, C., Computing the primary decomposition of zero-dimensional ideals, J. Symb. Comp. 34 (2002 Google Scholar), 451–459.

Morais, J.E., Resolución eficaz de systemas de ecuaciones polinomiales, Ph. D. thesis, University of Cantabria, Santander (1997 Google Scholar).

Moreno Maza, M., Rioboo, R., Polynomial gcd computation over tower of algebraic extension, in L. N. Comp. Sci. 948 (1995 Google Scholar), pp. 365–382, Springer.

Moritzugu, S., Kuriyama, K., On multiple zeros of systems of algebraic equations, in Proc. ISSAC'99 (1999 Google Scholar), pp. 23–30, Association for Computing Machinery.

Mourrain, B., Computing the isolated roots by matrix methods, J. Symb. Comp. 26 (1998 Google Scholar), 715–738.

Mourrain, B., A new criterion for normal form algorithms, in L. N. Comp. Sci. 1719 (1999 Google Scholar), pp. 430–443, Springer.

Mourrain, B., Bezoutian and quotient ring structure, J. Symb. Comp. 39 (2005 Google Scholar), 397–415.

Mourrain, B., Pan, Y.V., Multivariate polynomials, duality and structured matrices, J. Complexity 16 (2000 Google Scholar), 110–180.

Mourrain, B., Ruatta, O., Relation between roots and coefficients, interpolation and application to system solving, J. Symb. Comp. 33 (2002 Google Scholar), 679–699.

Mourrain, B., Trebuchet, P., Solving projective complete intersection faster, in Proc. ISSAC'00 (2000 Google Scholar), pp. 234–241, Association for Computing Machinery.

Muir, T., The Theory of Determinants in the Historical Order of Development, MacMillan (1906 Google Scholar).

Netto, E., Vorlesungen über Algebra, Zweiter Band Teubner (1900 Google Scholar).

Pardo, L.M., How lower and upper complexity bounds meet in elimination, in L. N. Comp. Sci. 948 (1995 Google Scholar), pp. 33–69, Springer.

Pierce, R.S., Modules over commutaive regular rings, Mem. A.M.S. 70 (1967 Google Scholar).

Pohst, M., Yun, D., On solving systems of algebraic equations via ideal bases and elimination, in Proc. 1981 SymSAC (1981 Google Scholar), pp. 206–211, Association for Computing Machinery.

Poisson, S.D.Mémoire sur l'élimination dans les équations algébriques, J. École Polytechnique t. IV (1802 Google Scholar), 199–203.

Rennert, N., Valibouze, A., Calcule de résolventes avec les modules de Cauchy, Exp. Math. 8 (1999 Google Scholar), 351–366.

Riordan, J., Combinatorial Identities, Wiley, (1968 Google Scholar).

Ritt, J.F., Prime and composite polynomials, Trans. A.M.S. 23 (1922 Google Scholar), 51–366.

Ritt, J.F., Differential Equations from the Algebraic Standpoint, A.M.S. Colloquium Publications 14 (1932 Google Scholar).

Ritt, J.F., Differential Algebra, A.M.S. Colloquium Publications 33 (1950 Google Scholar).

Robinson, L.B., Sur les systémes d'équations aux dérivées partialles, C.R. Acad. Sci. Paris 157 (1913 Google Scholar), 106–108.

Rouillier, F., Algorithmes efficaces pour l'étude des zéros réels des systèmes polynomiaux, Ph.D. thesis, University of Rennes I (1996 Google Scholar).

Rouillier, F., Solving zero-dimensional systems through the Rational Univariate Representation, J. AAECC 9 (1999 Google Scholar), 433–461.

Salmon, G., Lessons Introductory to the Modern Higher Algebra, Fifth Edn., Chelsea (1885 Google Scholar).

Sims, C., Computation with Finitely Presented Groups, Cambridge University Press (1994 Google Scholar).

Stetter, H., Matrix eigenprobelms are at the heart of polynomial system solving, SIGSAM Bull. 30 (1996 Google Scholar), 22–25.

Stetter, H., Numerical Polynomial Algebra, Tutorial Notes at ISSAC'98, Rostock (1998 Google Scholar).

Stetter, H., Numerical Polynomial Algebra, SIAM (2004 Google Scholar).

Sylvester, J.J., A method of determining by mere inspection the derivatives from two equations of any degree, Phil. Mag. XVI (1840 Google Scholar), 132–135.

Sylvester, J.J., Memoir on the dialytic method of elimination. Part I.Phil. Mag. XXXI (1842 Google Scholar), 534–539.

Sylvester, J.J., On a theory of the syzygietic relations of two rational integral functions, comprising an application to the theory of Sturm's functions, and that of the greatest algebraic common measure, Phil. Trans. Royal Soc. London CXLIII (1853 Google Scholar), 407–548.

Trinks, W., über, B.Buchberger Verfahren, Systeme algebraischer Gleichungen zu lösen, J. Numb. Theory 10 (1978 Google Scholar), 475–488.

Valibouze, A., Resolutions et functions symmetriques, in Proc. ISSAC'89 (1989 Google Scholar), pp. 390–399, Association for Computing Machinery.

Valibouze, A., Computation of the Galois groups of the resolvent factors for the direct and inverse Galois problems, in L. N. Comp. Sci. 948 (1995 Google Scholar), pp. 456–468, Springer.

Valibouze, A., Étude des relations algébriques entre les racines d'un polynôme d'une variable, Bull. Belg. Math. Soc. Simon Stevin 6 (1999 Google Scholar), 507–535.

Valibouze, A., Théorie de Galois constructive, Mémoir d'Habilitation, Paris (1998 Google Scholar).

Wang, D.-M., An elimination method for polynomial systems, J. Symb. Comp. 16 (1993 Google Scholar), 83–114.

Wang, D.-M., Decomposing polynomial systems into simple systems, J. Symb. Comp. 25 (1998 Google Scholar), 295–314.

Wimmer, H.K., On the history of the Bezoutian and the resultant matrix, Linear Algebra Appl. 128 (1990 Google Scholar), 27–34.

Wu, W.-T., On the decision problem and the mechanization of the theorem-proving in elementary geometry, Scinetia Sinica 21 (1978 Google Scholar), 159–172.

Wu, W.-T., Basic principles of mechanical theorem proving in elementary geometry, J. Sys. Sci. & Math. Sci. 4 (1984 Google Scholar), 207–235.

Wu, W.-T., On the decision problem and the mechanization of the theorem-proving in elementary geometry, in Contemp. Math. 29 (1984 Google Scholar), pp. 213–234, American Mathematical Society.

Wu, W.-T., Some recent advances in mechanical theorem-proving of geometry, (reprinted) in Contemp. Math. 29 (1984 Google Scholar), pp. 235–241, American Mathematical Society.

Wu, W.-T., A zero structure theorem for polynomial equations solving, M.M. Research Preprints 1 (1987 Google Scholar), 2–12.

Wißmann, D., Anwendung von Rewriting-Techniken in polyzyklischen Gruppen, Dissertation, Kaiserslautern (1989 Google Scholar).

Yokoyama, K., Noro, M., Takeshima, T., Solutions of systems of algebraic equations and linear maps on residue class rings, J. Symb. Comp. 14 (1992 Google Scholar), 399–417.

Zariski, O., Samuel, P., Commutative Algebra, Van Nostrand (1958 Google Scholar).

Solving Polynomial Equation Systems III

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Contents

Frontmatter
pp i-iv

Dedication
pp v-vi

Contents
pp vii-x

Preface
pp xi-xii

Setting
pp xiii-xviii

PART SIX - Algebraic Solving
pp 1-2

39 - Trinks
pp 3-18

40 - Stetter
pp 19-65

41 - Macaulay IV
pp 66-137

42 - Lazard II
pp 138-191

43 - Lagrange II
pp 192-221

44 - Kronecker IV
pp 222-246

45 - Duval II
pp 247-266

Bibliography
pp 267-272

Index
pp 273-275

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