Details:
Title  Pivoting in extended rings for computing approximate Gr\"obner bases.  Author(s)  Faug`ere JeanCharles, Ye Liang  Type  Article in Journal  Abstract  It is well known that in the computation of Gröbner bases arbitrarily small perturbations in the coefficients of polynomials may lead to a completely different staircase, even if the solutions of the polynomial system change continuously. This phenomenon is called artificial discontinuity in Kondratyev’s Ph.D. thesis. We show how such phenomenon may be detected and even “repaired” by using a new variable to rename the leading term each time we detect a “problem”. We call such strategy the TSV (Term Substitutions with Variables) strategy. For a zerodimensional polynomial ideal, any monomial basis (containing 1) of the quotient ring can be found with the TSV strategy. Hence we can use TSV strategy to relax term order while keeping the framework of Gröbner basis method so that we can use existing efficient algorithms (for instance the F 5 algorithm) to compute an approximate Gröbner basis. Our main algorithms, named TSVn and TSVh, can be used to repair artificial ϵdiscontinuities. Experiments show that these algorithms are effective for some nontrivial problems.  Keywords  Approximate Gröbner basis, Artificial discontinuity, Monomial basis, F 5 algorithm  ISSN  16618270; 16618289/e 
URL 
http://link.springer.com/article/10.1007%2Fs117860110089y 
Language  English  Journal  Math. Comput. Sci.  Volume  5  Number  2  Pages  179194  Publisher  Springer (Birkh\"auser), Basel  Year  2011  Edition  0  Translation 
No  Refereed 
No 
