Abstract:
Let A be a polynomial ring in n+1 variables over an arbitrary infinite field k. It is proved that for all sufficiently large n and d there is a homogeneous prime ideal p⊂A satisfying the following conditions. The ideal p corresponds to a component, defined over k and irreducible over ¯¯¯k, of a projective algebraic variety given by a system of homogeneous polynomial equations with polynomials in A of degrees less than d. Any system of generators of p contains a polynomial of degree at least d2cn for an absolute constant c>0, which can be computed efficiently. This solves an important old problem in effective algebraic geometry. For the case of finite fields a slightly less strong result is obtained.
Citation:
A. L. Chistov, “Double-exponential lower bound for the degree of any system of generators of a polynomial prime ideal”, Algebra i Analiz, 20:6 (2008), 186–213; St. Petersburg Math. J., 20:6 (2009), 983–1001
\Bibitem{Chi08}
\by A.~L.~Chistov
\paper Double-exponential lower bound for the degree of any system of generators of a~polynomial prime ideal
\jour Algebra i Analiz
\yr 2008
\vol 20
\issue 6
\pages 186--213
\mathnet{http://mi.mathnet.ru/aa544}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2530898}
\zmath{https://zbmath.org/?q=an:1206.13031}
\transl
\jour St. Petersburg Math. J.
\yr 2009
\vol 20
\issue 6
\pages 983--1001
\crossref{https://doi.org/10.1090/S1061-0022-09-01081-4}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000272556200006}
Linking options:
https://www.mathnet.ru/eng/aa544
https://www.mathnet.ru/eng/aa/v20/i6/p186
This publication is cited in the following 12 articles:
M. V. Kondratieva, “Generalized Typical Dimension of a Graded Module”, J Math Sci, 262:5 (2022), 691
M. V. Kondratieva, “A Bound for a Typical Differential Dimension of Systems of Linear Differential Equations”, J Math Sci, 259:4 (2021), 563
Simmons W., Towsner H., “Explicit Polynomial Bounds on Prime Ideals in Polynomial Rings Over Fields”, Pac. J. Math., 306:2 (2020), 721–754
A. L. Chistov, “Vychislenie izolirovannykh primarnykh komponent polinomialnogo ideala za subeksponentsialnoe vremya”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XXXI, Zap. nauchn. sem. POMI, 498, POMI, SPb., 2020, 64–74
Amzallag E., Sun M., Pogudin G., Vo T.N., “Complexity of Triangular Representations of Algebraic Sets”, J. Algebra, 523 (2019), 342–364
Davenport J.H., “The Role of Benchmarking in Symbolic Computation (Position Paper)”, 2018 20Th International Symposium on Symbolic and Numeric Algorithms For Scientific Computing (Synasc 2018), International Symposium on Symbolic and Numeric Algorithms For Scientific Computing, IEEE Computer Soc, 2019, 275–279
Davenport J., “Methodologies of Symbolic Computation”, Artificial Intelligence and Symbolic Computation (Aisc 2018), Lecture Notes in Artificial Intelligence, 11110, eds. Fleuriot J., Wang D., Calmet J., Springer International Publishing Ag, 2018, 19–33
A. L. Chistov, “Estimating the power of a system of equations that determines a variety of reducible polynomials”, St. Petersburg Math. J., 24:3 (2013), 513–528
A. L. Chistov, “An effective version of the first Bertini theorem in nonzero characteristic and its applications”, J. Math. Sci. (N. Y.), 190:3 (2013), 503–514
Gerdt V.P., Zinin M.V., Blinkov Yu.A., “On computation of Boolean involutive bases”, Programming and Computer Software, 36:2 (2010), 117–123
Chistov A. L., “Effective normalization of a nonsingular in codimension one algebraic variety”, Dokl. Math., 80:1 (2009), 577–580