A Gröbner free alternative for polynomial system solving
Journal of Complexity
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
Semidefinite representations for finite varieties
Mathematical Programming: Series A and B
Moment matrices, trace matrices and the radical of ideals
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals
Foundations of Computational Mathematics
Multihomogeneous polynomial decomposition using moment matrices
Proceedings of the 36th international symposium on Symbolic and algebraic computation
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In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming its complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of Mourrain and Trebuchet (2005) are efficient and numerically stable for computing complex roots, algorithms based on moment matrices (Lasserre et al., 2008) allow the incorporation of additional polynomials, e.g., to restrict the computation to real roots or to eliminate multiple solutions. The proposed algorithm can be used to compute a border basis of the input ideal and, as opposed to other approaches, it can also compute the quotient structure of the (real) radical ideal directly, i.e., without prior algebraic techniques such as Grobner bases. It thus combines the strength of existing algorithms and provides a unified treatment for the computation of border bases for the ideal, the radical ideal and the real radical ideal.