A New Criterion for Normal Form Algorithms
AAECC-13 Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Generalized normal forms and polynomial system solving
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Border bases of positive dimensional polynomial ideals
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals
Foundations of Computational Mathematics
Stable normal forms for polynomial system solving
Theoretical Computer Science
Numerical calculation of H-bases for positive dimensional varieties
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
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In this paper, we generalized the construction of border bases to non-zero dimensional ideals for normal forms compatible with the degree, tackling the remaining obstacle for a general application of border basis methods. First, we give conditions to have a border basis up to a given degree. Next, we describe a new stopping criteria to determine when the reduction with respect to the leading terms is a normal form. This test based on the persistence and regularity theorems of Gotzmann yields a new algorithm for computing a border basis of any ideal, which proceeds incrementally degree by degree until its regularity. We detail it, prove its correctness, present its implementation and report some experimentations which illustrate its practical good behavior.