The 40 “generic” positions of a parallel robot
ISSAC '93 Proceedings of the 1993 international symposium on Symbolic and algebraic computation
Efficient computation of zero-dimensional Gro¨bner bases by change of ordering
Journal of Symbolic Computation
A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Computing the isolated roots by matrix methods
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Matrices in elimination theory
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
Solving projective complete intersection faster
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
A Gröbner free alternative for polynomial system solving
Journal of Complexity
Numerical stability and stabilization of Groebner basis computation
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Numerical Polynomial Algebra
Generalized normal forms and polynomial system solving
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Border basis detection is NP-complete
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Flat families by strongly stable ideals and a generalization of Gröbner bases
Journal of Symbolic Computation
Complexity of Gröbner basis detection and border basis detection
Theoretical Computer Science
Border basis representation of a general quotient algebra
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Hi-index | 5.23 |
The paper describes and analyzes a method for computing border bases of a zero-dimensional ideal I. The criterion used in the computation involves specific commutation polynomials, and leads to an algorithm and an implementation extending the ones in [B. Mourrain, Ph. Trebuchet, Generalised normal forms and polynomial system solving, in: M. Kauers (Ed.), Proc. Intern. Symp. on Symbolic and Algebraic Computation, ACM Press, New-York, 2005, pp. 253-260]. This general border basis algorithm weakens the monomial ordering requirement for Grobner bases computations. It is currently the most general setting for representing quotient algebras, embedding into a single formalism Grobner bases, Macaulay bases and a new representation that does not fit into the previous categories. With this formalism, we show how the syzygies of the border basis are generated by commutation relations. We also show that our construction of normal form is stable under small perturbations of the ideal, if the number of solutions remains constant. This feature has a huge impact on practical efficiency, as illustrated by the experiments on classical benchmark polynomial systems, at the end of the paper.