Degree bounds for Gröbner bases of low-dimensional polynomial ideals

Let K[<i>X</i>] be a ring of multivariate polynomials with coefficients in a field K, and let <i>f</i><sub>1</sub>, ..., <i>f</i><sub><i>s</i></sub> be polynomials with maximal total degree <i>d</i> which generate an ideal <i>I</i> of dimension <i>r</i>. Then, for every admissible ordering, the total degree of polynomials in a Gröbner basis for <i>I</i> is bounded by 2 (1/2<i>d</i><sup><i>n-r</i></sup> + <i>d</i>)<sup>2</sup><sup><i>r</i></sup>. This is proved using the cone decompositions introduced by Dubé in [5]. Also, a lower bound of similar form is given.

[1]  임종인,et al.  Gröbner Bases와 응용 , 1995 .

[2]  Daniel Lazard,et al.  Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations , 1983, EUROCAL.

[3]  Thomas Dubé,et al.  The Structure of Polynomial Ideals and Gröbner Bases , 2013, SIAM J. Comput..

[4]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[5]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[6]  Martin Kreuzer,et al.  Computational Commutative Algebra 1 , 2000 .

[7]  Chee Yap.  A new lower bound construction for commutative Thue systems with applications: RISC Linz Technical Report No. 88-28.0 , 1988 .

[8]  Marc Giusti,et al.  Some Effectivity Problems in Polynomial Ideal Theory , 1984, EUROSAM.

[9]  Joachim Schmid,et al.  On the affine Bezout inequality , 1995 .

[10]  I. Shafarevich.  Basic algebraic geometry , 1974 .

[11]  Martin Raab,et al.  Computing the Dimension of a Polynomial Ideal , 2007 .

[12]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[13]  A. Meyer,et al.  The complexity of the word problems for commutative semigroups and polynomial ideals , 1982 .

[14]  Hans Schönemann,et al.  SINGULAR: a computer algebra system for polynomial computations , 2001, ACCA.

[15]  H. Michael Möller,et al.  Upper and Lower Bounds for the Degree of Groebner Bases , 1984, EUROSAM.

[16]  D. Bayer.  The division algorithm and the hilbert scheme , 1982 .

[17]  Ernst W. Mayr,et al.  Exponential space computation of Gröbner bases , 1996, ISSAC '96.