Consider a system of polynomial equations in n variables of degrees at most d with the set of all common zeros V. We suggest subexponential time algorithms (in the general case and in the case of zero characteristic) for constructing n+1 equations of degrees at most d defining the algebraic variety V.
Further, we construct n equations defining V. We give an explicit upper bound on the degrees of these n equations. It is double exponential in n. The running time of the algorithm for constructing them is also double exponential in n.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. L. Chistov, “Polynomial complexity algorithm for factoring polynomials and constructing components of a variety in subexponential time,” Zap. Nauchn. Semin. POMI, 137, 124–188 (1984).
A. L. Chistov, “An improvement of the complexity bound for solving systems of polynomial equations,” Zap. Nauchn. Semin. POMI, 390, 299–306 (2011).
A. L. Chistov, “Systems with parameters, or efficiently solving systems of polynomial equations: 33 years later. I,” Zap. Nauchn. Semin. POMI, 462, 122–166 (2017).
A. L. Chistov, “Systems with parameters, or efficiently solving systems of polynomial equations: 33 years later. II,” Zap. Nauchn. Semin. POMI, 468, 138–176 (2018).
A. L. Chistov, “Polynomial-time algorithms for a new model of representation of algebraic varieties (in characteristic zero),” Zap. Nauchn. Semin. POMI, 378, 133–170 (2010).
A. L. Chistov, “Efficient absolute factorization of polynomials with parametric coefficients,” Zap. Nauchn. Semin. POMI, 448, 286–325 (2016).
G. E. Collins, “Polynomial remainder sequences and determinants,” Amer. Math. Monthly, 73, No. 7, 708–712 (1966).
L. Kronecker, “Grundzüge einer arithmetischen Theorie der algebraischen Grössen,” J. reine angew. Math., 92, 1–123 (1882).
U. Storch, “Bemerkung zu einem Satz von M. Kneser,” Arch. Math., 23, 403–404 (1972).
D. Eisenbud and E. G. Evans, Jr, “Every algebraic set in n-space is the intersection of n hypersurfaces,” Inv. Math., 19, 107–112 (1973).
E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser (1985).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 507, 2021, pp. 140–156.
Rights and permissions
About this article
Cite this article
Chistov, A.L. An Effective Construction of a Small Number of Equations Defining an Algebraic Variety. J Math Sci 261, 687–697 (2022). https://doi.org/10.1007/s10958-022-05780-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-05780-5