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Algebra i Analiz, 2008, Volume 20, Issue 6, Pages 186–213 (Mi aa544)  

This article is cited in 12 scientific papers (total in 12 papers)

Research Papers

Double-exponential lower bound for the degree of any system of generators of a polynomial prime ideal

A. L. Chistov

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
References:
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 pA 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.
Keywords: Polynomial ideal, projective algebraic variety, Gröbner basis, effective algebraic geometry.
Received: 10.04.2008
English version:
St. Petersburg Mathematical Journal, 2009, Volume 20, Issue 6, Pages 983–1001
DOI: https://doi.org/10.1090/S1061-0022-09-01081-4
Bibliographic databases:
Document Type: Article
MSC: 13P10, 14Q20
Language: Russian
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
Citation in format AMSBIB
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\jour Algebra i Analiz
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\vol 20
\issue 6
\pages 186--213
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\transl
\jour St. Petersburg Math. J.
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\vol 20
\issue 6
\pages 983--1001
\crossref{https://doi.org/10.1090/S1061-0022-09-01081-4}
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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:
    1. M. V. Kondratieva, “Generalized Typical Dimension of a Graded Module”, J Math Sci, 262:5 (2022), 691  crossref
    2. M. V. Kondratieva, “A Bound for a Typical Differential Dimension of Systems of Linear Differential Equations”, J Math Sci, 259:4 (2021), 563  crossref
    3. Simmons W., Towsner H., “Explicit Polynomial Bounds on Prime Ideals in Polynomial Rings Over Fields”, Pac. J. Math., 306:2 (2020), 721–754  crossref  mathscinet  isi  scopus
    4. 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  mathnet
    5. Amzallag E., Sun M., Pogudin G., Vo T.N., “Complexity of Triangular Representations of Algebraic Sets”, J. Algebra, 523 (2019), 342–364  crossref  mathscinet  zmath  isi  scopus
    6. 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  crossref  isi  scopus
    7. 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  crossref  isi  scopus
    8. 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  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    9. 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  mathnet  crossref  mathscinet
    10. Gerdt V.P., Zinin M.V., Blinkov Yu.A., “On computation of Boolean involutive bases”, Programming and Computer Software, 36:2 (2010), 117–123  crossref  mathscinet  zmath  isi  elib  scopus
    11. Chistov A. L., “Effective normalization of a nonsingular in codimension one algebraic variety”, Dokl. Math., 80:1 (2009), 577–580  mathnet  crossref  mathscinet  mathscinet  zmath  isi  elib  elib
    12. J. Math. Sci. (N. Y.), 168:3 (2010), 478–490  mathnet  crossref
    Citing articles in Google Scholar: Russian citations, English citations
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