Abstract
Let k be a ground field of zero characteristic, and let V be an algebraic variety over k given as the locus of a family of polynomials of degree less than d in n variables. In the paper, we construct algorithms that have working time that is polynomial in the size of the input and d n and compute the following: the degree of the variety V, the dimension of V in a neighborhood of a given point, the multiplicity of a given point of V, and a representative system of smooth points with their tangent spaces on each component of V. Also, we construct an algorithm for deciding whether a given morphism between two given algebraic varieties V and V' is dominant. Bibliography: 17 titles.
Similar content being viewed by others
REFERENCES
J. Bochnak, M. Coste, and M.-F. Roy, Géométrie Algébrique Réelle, Springer-Verlag, Berlin-Heidelberg-New York (1987).
A. L. Chistov, "Polynomial-time computation of the dimension of algebraic varieties in zero characteristic," J. Symb. Comp., 22, 1-25 (1996).
A. L. Chistov, "Polynomial-time computation of the dimensions of components of algebraic varieties in zero characteristic," J. Pure Appl. Algebra, 117, 118, 145-175 (1997).
A. L. Chistov, "A polynomial-complexity algorithm for factoring polynomials and constructing components of a variety in subexponential time," Zap. Nauchn. Semin. LOMI, 137, 124-188 (1984).
A. L. Chistov, "Polynomial complexity of the Newton-Puiseux algorithm," Lect. Notes Comp. Sci., 233, 247-255 (1986).
A. L. Chistov, "Polynomial complexity algorithms for computational problems in the theory of algebraic curves," Zap. Nauchn. Semin. LOMI, 176, 127-150 (1989).
A. L. Chistov and M. Karpinski, "Complexity of deciding solvability of polynomial equations over p-adics," Research Report No. 85183-CS, Inst. Informatik, Univ. Bonn (1997).
M. Giusti and J. Heintz, "La détermination des points isolés et de la dimension d'une variété algébrique peut se faire en temps polynomial," in: Proc. Int. Meeting Comm. Algebra, Cortona (1993).
R. Hartshorne, Algebraic Geometry, Springer-Verlag, Berlin-Heidelberg-New York (1977).
J. Igusa, Mem. Coll. Sci. Univ. Kyoto, 27, 189-201 (1951).
L. Kroneker, "Grundzüge einer arithmetischen Theorie der algebraischen Grossen," J. Reine Angew. Math., 92, 1-122 (1882).
D. Lazard, "Algèbre linéaire sur k[x1;:::; xn] et élimination," Bull. Soc. Math. France, 105, 165-190 (1977).
D. Lazard, "Résolution des systémes d'équations algébrique," Theor. Comp. Sci., 15, 77-110 (1981).
D. Mumford, Algebraic Geometry I. Complex Projective Varieties, Springer-Verlag, Berlin-Heidelberg-New York (1976).
J. Milnor, "On Betti numbers of real varieties," Proc. Am. Math. Soc., 15 (2), 275-280 (1964).
J. Renegar, "A faster PSPACE algorithm for deciding the existential theory of reals," in: Proc. 29th Annual Symp. Found. Comp. Sci. (1988), pp. 291-295.
P. Samuel, Métodes d'Algebre Abstracte en Géométrie Algébrique (1967).
Rights and permissions
About this article
Cite this article
Chistov, A.L. Polynomial-Time Computation of the Degree of Algebraic Varieties in Zero Characteristic and Its Applications. Journal of Mathematical Sciences 108, 897–933 (2002). https://doi.org/10.1023/A:1013527102517
Issue Date:
DOI: https://doi.org/10.1023/A:1013527102517