Computing the Radical of an Ideal in Positive Characteristic

We propose a method for computing the radical of an arbitrary ideal in the polynomial ring in n variables over a perfect field of characteristic p 0. In our method Buchberger?s algorithm is performed once in n variables and a Grobner basis conversion algorithm is performed at most ?nlogpd? times in 2 n variables, where d is the maximum of total degrees of generators of the ideal and 3. Next we explain how to compute radicals over a finitely generated coefficient field over a field K, when we have a radical computation method over the field K. Thus we can compute radicals over any finitely generated field over a perfect field.

[1]  CARLO TRAVERSO,et al.  Hilbert Functions and the Buchberger Algorithm , 1996, J. Symb. Comput..

[2]  Teresa Krick,et al.  An Algorithm for the Computation of the Radical of an Ideal in the Ring of Polynomials , 1991, AAECC.

[3]  Wu Wen-tsun.  Basic principles of mechanical theorem proving in elementary geometries , 1986 .

[4]  G. Greuel,et al.  SINGULAR version 1.2 User Manual , 1998 .

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

[6]  Dongming Wang,et al.  An Elimination Method for Polynomial Systems , 1993, J. Symb. Comput..

[7]  Dongming Wang.  Unmixed and prime decomposition of radicals of polynomial ideals , 1998, SIGS.

[8]  D. Eisenbud,et al.  Direct methods for primary decomposition , 1992 .

[9]  Pablo Solernó,et al.  On the Computation of the Radical of Polynomial Complete Intersection Ideals , 1995, AAECC.

[10]  Patrizia M. Gianni,et al.  Gröbner Bases and Primary Decomposition of Polynomial Ideals , 1988, J. Symb. Comput..

[11]  Quoc-Nam Tran.  A Fast Algorithm for Gröbner Basis Conversion and its Applications , 2000, J. Symb. Comput..

[12]  Michael Kalkbrener,et al.  Converting Bases with the Gröbner Walk , 1997, J. Symb. Comput..

[13]  L. O'carroll.  AN INTRODUCTION TO GRÖBNER BASES (Graduate Studies in Mathematics 3) , 1996 .

[14]  Wolfgang Küchlin,et al.  On the Walk , 1997, Theor. Comput. Sci..

[15]  Jean-Charles Faugère,et al.  Efficient Computation of Zero-Dimensional Gröbner Bases by Change of Ordering , 1993, J. Symb. Comput..

[16]  Wolmer V. Vasconcelos,et al.  Computational methods in commutative algebra and algebraic geometry , 1997, Algorithms and computation in mathematics.