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Keywords:
Geometry of polar varieties and its generalizations; geometric degree; real polynomial equation solving; elimination procedure; arithmetic circuit; arithmetic network; complexity
Geometry of polar varieties and its generalizations; geometric degree; real polynomial equation solving; elimination procedure; arithmetic circuit; arithmetic network; complexity
Summary:
Let $W$ be a closed algebraic subvariety of the $n$-dimensional projective space over the complex or real numbers and suppose that $W$ is non-empty and equidimensional. In this paper we generalize the classic notion of polar variety of $W$ associated with a given linear subvariety of the ambient space of $W$. As particular instances of this new notion of generalized polar variety we reobtain the classic ones and two new types of polar varieties, called dual and (in case that $W$ is affine) conic. We show that for a generic choice of their parameters the generalized polar varieties of $W$ are empty or equidimensional and, if $W$ is smooth, that their ideals of definition are Cohen-Macaulay. In the case that the variety $W$ is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of $W$ by explicit equations. Finally, we use this description in order to design a new, highly efficient elimination procedure for the following algorithmic task: In case, that the variety $W$ is $\mathbb{Q}$-definable and affine, having a complete intersection ideal of definition, and that the real trace of $W$ is non-empty and smooth, find for each connected component of the real trace of $W$ a representative point.
References:
[1] Aubry P., Rouillier, F., Din M. Safey El: Real solving for positive dimensional systems. J. Symb. Computation 34 (2002), 6, 543–560 DOI 10.1006/jsco.2002.0563 | MR 1943042
[2] Bank B., Giusti M., Heintz, J., Pardo L. M.: Generalized polar varieties: Geometry and algorithms. Manuscript. Humboldt Universität zu Berlin, Institut für Mathematik 2003. Submitted to J. Complexity MR 2152713 | Zbl 1085.14047
[4] Bank B., Giusti M., Heintz, J., Mbakop G. M.: Polar varieties and efficient real elimination. Math.Z. 238 (2001), 115–144. Digital Object Identifier (DOI) 10.1007/s002090100248 DOI 10.1007/PL00004896 | MR 1860738 | Zbl 1073.14554
[5] Bank B., Giusti M., Heintz, J., Mbakop G. M.: Polar varieties, real equation solving and data structures: The hypersurface case. J. Complexity 13 (1997), 1, 5–27. Best Paper Award J. Complexity 1997 DOI 10.1006/jcom.1997.0432 | MR 1449757 | Zbl 0872.68066
[6] Basu S., Pollack, R., Roy M.-F.: On the combinatorial and algebraic complexity of quantifier elimination. J.ACM 43 (1996), 6, 1002–1045 DOI 10.1145/235809.235813 | MR 1434910 | Zbl 0885.68070
[7] Basu S., Pollack, R., Roy M.-F.: Complexity of computing semi-algebraic descriptions of the connected components of a semialgebraic set. In: Proc. ISSAC’98, (XXX. Gloor and XXX. Oliver, eds.), Rostock 1998, pp. 13–15. ACM Press. (1998), 25–29 MR 1805168 | Zbl 0960.14033
[8] Bochnak J., Coste, M., Roy M.–F.: Géométrie algébrique réelle. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin – Heidelberg – New York 1987 MR 0949442 | Zbl 0633.14016
[9] Bruns W., Vetter U.: Determinantal Rings. (Lecture Notes in Mathematics 1327), Springer, Berlin 1988 MR 0953963 | Zbl 1079.14533
[10] Bürgisser P., Clausen, M., Shokrollahi M. A.: Algebraic complexity theory. With the collaboration of Thomas Lickteig. (Grundlehren der Mathematischen Wissenschaften 315), Springer, Berlin 1997 MR 1440179 | Zbl 1087.68568
[11] Canny J. F.: Some algebraic and geometric computations in PSPACE. In: Proc. 20th ACM Symp. on Theory of Computing 1988, pp. 460–467
[12] Canny J. F., Emiris I. Z.: Efficient incremental algorithms for the sparse resultant and the mixed volume. J. Symb. Comput. 20 (1995), 2, 117–149 DOI 10.1006/jsco.1995.1041 | MR 1374227 | Zbl 0843.68036
[13] Castro D., Giusti M., Heintz J., Matera, G., Pardo L. M.: The hardness of polynomial solving. Found. Comput. Math. 3, (2003), 4, 347–420 DOI 10.1007/s10208-002-0065-7 | MR 2009683
[14] Castro D., Hägele K., Morais J. E., Pardo L. M.: Kronecker’s and Newton’s approaches to solving: A first comparision. J. Complexity 17 (2001), 1, 212–303 DOI 10.1006/jcom.2000.0572 | MR 1817613
[15] Coste M., Roy M.-F.: Thom’s Lemma, the coding of real algebraic numbers and the computation of the topology of semialgebraic sets. J. Symbolic Comput. 5 (1988), 121–130 DOI 10.1016/S0747-7171(88)80008-7 | MR 0949115
[16] Cucker F., Smale S.: Complexity estimates depending on condition and round-of error. J. ACM 46 (1999), 1, 113–184 DOI 10.1145/300515.300519 | MR 1692497
[17] Demazure M.: Catastrophes et bifurcations. Ellipses, Paris 1989 Zbl 0907.58002
[18] Eagon J. A., Northcott D. G.: Ideals defined by matrices and a certain complex associated with them. Proc. R. Soc. Lond., Ser. A 269 (1962), 188–204 DOI 10.1098/rspa.1962.0170 | MR 0142592 | Zbl 0106.25603
[19] Eagon J. A., Hochster M.: R–sequences and indeterminates. O.J. Math., Oxf. II. Ser. 25 (1974), 61–71 MR 0337934 | Zbl 0278.13008
[20] Eisenbud D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, New York 1995 MR 1322960 | Zbl 0819.13001
[21] Fulton W.: Intersection Theory. (Ergebnisse der Mathematik und ihrer Grenzgebiete 3), Springer, Berlin 1984 MR 0732620 | Zbl 0935.00036
[22] Gathen J. von zur: Parallel linear algebra. In: Synthesis of Parallel Algorithmns (J. Reif, ed.), Morgan Kaufmann 1993
[23] Giusti M., Heintz J.: Kronecker’s smart, little black boxes. Foundations of Computational Mathematics (R. A. DeVore, A. Iserles, and E. Süli, eds.), Cambridge Univ. Press 284 (2001), 69–104 DOI 10.1017/CBO9781107360198.005 | MR 1836615 | Zbl 0978.65043
[24] Giusti M., Heintz J., Morais J. E., Morgenstern, J., Pardo L. M.: Straight–line programs in geometric elimination theory. J. Pure Appl. Algebra 124 (1998), 1 – 3, 101–146 DOI 10.1016/S0022-4049(96)00099-0 | MR 1600277 | Zbl 0944.12004
[25] Giusti M., Heintz J., Hägele K., Morais J. E., Montaña J. L., Pardo L. M.: Lower bounds for diophantine approximations. J. Pure and Applied Alg. 117, 118 (1997), 277–317 DOI 10.1016/S0022-4049(97)00015-7 | MR 1457843 | Zbl 0871.68101
[26] Giusti M., Lecerf, G., Salvy B.: A Gröbner free alternative for polynomial system solving. J. Complexity 17 (2001), 1, 154–211 DOI 10.1006/jcom.2000.0571 | MR 1817612 | Zbl 1003.12005
[27] Grigor’ev D., Vorobjov N.: Solving systems of polynomial inequalities in subexponential time. J. Symb. Comput. 5 (1998), 1/2, 37–64 DOI 10.1016/S0747-7171(88)80005-1 | MR 0949112
[28] Heintz J.: Fast quantifier elimination over algebraically closed fields. Theoret. Comp. Sci. 24 (1983), 239–277 DOI 10.1016/0304-3975(83)90002-6 | MR 0716823
[29] Heintz J., Matera, G., Waissbein A.: On the time–space complexity of geometric elimination procedures. Appl. Algebra Engrg. Comm. Comput. 11 (2001), 4, 239–296 DOI 10.1007/s002000000046 | MR 1818975 | Zbl 0977.68101
[31] Heintz J., Roy, M.–F., Solernó P.: Complexité du principe de Tarski–Seidenberg. CRAS, t. 309, Série I, Paris 1989, pp. 825–830 MR 1055203 | Zbl 0704.03013
[32] Heintz J., Roy,, M–F., Solernó P.: Sur la complexité du principe de Tarski–Seidenberg. Bull. Soc. Math. France 18 (1990), 101–126 MR 1077090 | Zbl 0767.03017
[33] Heintz J., Schnorr C. P.: Testing polynomials which are easy to compute. In: Proc. 12th Ann. ACM Symp. on Computing, 1980, pp. 262–268, also in: Logic and Algorithmic: An Int. Symposium held in Honour of Ernst Specker, Monographie No.30 de l’Enseignement de Mathématiques, Genève 1982, pp. 237–254 MR 0648305
[34] Krick T., Pardo L. M.: A computational method for diophantine approximation. In: Proc. MEGA’94, Algorithms in Algebraic Geometry and Applications, (L. Gonzales-Vega and T. Recio, eds.), (Progress in Mathematics 143), Birkhäuser, Basel 1996, pp. 193-254 MR 1414452 | Zbl 0878.11043
[35] Kunz E.: Kähler Differentials. Advanced Lectures in Mathematics. Vieweg, Braunschweig – Wiesbaden 1986 MR 0864975 | Zbl 0587.13014
[36] Lê D. T., Teissier B.: Variétés polaires locales et classes de Chern des variétés singulières. Annals of Mathematics 114 (1981), 457–491 DOI 10.2307/1971299 | MR 0634426 | Zbl 0488.32004
[37] Lecerf G.: Une alternative aux méthodes de réécriture pour la résolution des systèmes algébriques. Thèse. École Polytechnique 2001
[38] Lecerf G.: Quadratic Newton iterations for systems with multiplicity. Found. Comput. Math. 2 (2002), 3, 247–293 DOI 10.1007/s102080010026 | MR 1907381
[39] Lehmann L., Waissbein A.: Wavelets and semi–algebraic sets. In: WAIT 2001, (M. Frias and J. Heintz eds.), Anales JAIIO 30 (2001), 139–155
[40] Matera G.: Probabilistic algorithms for geometric elimination. Appl. Algebra Engrg. Commun. Comput. 9 (1999), 6, 463–520 DOI 10.1007/s002000050115 | MR 1706433 | Zbl 0934.68122
[41] Matsumura H.: Commutative Ring Theory. (Cambridge Studies in Adv. Math. 8), Cambridge Univ. Press, Cambridge 1986 MR 0879273 | Zbl 0666.13002
[42] Piene R.: Polar classes of singular varieties. Ann. Scient. Éc. Norm. Sup. 4. Série, t. 11 (1978), 247–276 MR 0510551 | Zbl 0401.14007
[43] Renegar J.: A faster PSPACE algorithm for the existential theory of the reals. In: Proc. 29th Annual IEEE Symposium on the Foundation of Computer Science 1988, pp. 291–295
[44] Renegar J.: On the computational complexity and geometry of the first order theory of the reals. J. Symbolic Comput. 13 (1992), 3, 255–352 MR 1156882 | Zbl 0798.68073
[45] Din M. Safey El: Résolution réelle des systèmes polynomiaux en dimension positive. Thèse. Université Paris VI 2001
[46] Din M. Safey El, Schost E.: Polar varieties and computation of one point in each connected component of a smooth real algebraic set. In: Proc. 2003 International Symposium on Symbolic and Algebraic Computation (ISSAC 2003), ACM Press 2003, pp. 224–231 MR 2035216
[47] Spivak M.: Calculus on Manifolds. A Modern Approach to Classical Theorems of Calculus. W. A. Benjamin, Inc., New York – Amsterdam 1965 MR 0209411 | Zbl 0381.58003
[48] Teissier B.: Variétés Polaires. II. Multiplicités polaires, sections planes et conditions de Whitney. Algebraic geometry (La Rábida, 1981), (J. M. Aroca, R. Buchweitz, M. Giusti and M. Merle eds.), pp. 314–491, (Lecture Notes in Math. 961) Springer, Berlin 1982, pp. 314–491 MR 0708342 | Zbl 0572.14002
[49] Vogel W.: Lectures on Results on Bézout’s Theorem. Springer, Berlin 1984 MR 0743265 | Zbl 0553.14022
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