Moment matrices, border bases and real radical computation

@article{Lasserre2013MomentMB,
  title={Moment matrices, border bases and real radical computation},
  author={Jean Bernard Lasserre and Monique Laurent and Bernard Mourrain and Philipp Rostalski and Philippe Trebuchet},
  journal={J. Symb. Comput.},
  year={2013},
  volume={51},
  pages={63-85}
}
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28 Citations
Highly Influential Citations
1
Background Citations
9
Methods Citations
6
Results Citations
2

28 Citations

Computing Real Radicals by Moment Optimization
TLDR
A new algorithm for computing the real radical of an ideal I and, more generally, the S-radical of I is presented, which is based on convex moment optimization, and an effective, general stopping criterion is given on the degree to detect when the prime ideals lying over the annihilator are real.
The Canonical Decomposition of Cnd and Numerical Gröbner and Border Bases
TLDR
It is demonstrated how the canonical decomposition can be used to decide whether the affine solution set of a multivariate polynomial system is zero-dimensional and to solve the ideal membership problem numerically.
On exact Reznick, Hilbert-Artin and Putinar's representations
Border Basis for Polynomial System Solving and Optimization
TLDR
The software package borderbasix is described, dedicated to the computation of border bases and the solutions of polynomial equations, and the main ingredients of the border basis algorithm and the other methods implemented.
On the Complexity of Computing Real Radicals of Polynomial Systems
TLDR
A probabilistic algorithm of complexity sO(1) (nD)O(nr2r) to compute rational parametrizations for all irreducible components of the real algebraic set V ∩ Rn is given.
On Exact Polya, Hilbert-Artin and Putinar's Representations
TLDR
This work provides a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions for polynomials lying in the interior of the SOS cone, and proves that bit complexity estimates on output size and runtime are both polynomial in the degree of the inputPolynomial and simply exponential in the number of variables.
On Exact Polya and Putinar's Representations
TLDR
This work provides a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions for polynomials lying in the interior of the SOS cone and proves that bit complexity estimates on output size and runtime are both polynomial in the degree of the inputPolynomial and simply exponential in the number of variables.
A certificate for semidefinite relaxations in computing positive-dimensional real radical ideals
Multivariate moment problems : applications of the reconstruction of linear forms on the polynomial ring
This thesis deals with the reconstruction of linear forms on the polynomial ring and its applications. We propose theoretical and algorithmic tools to solve multivariate moment problems: the…
Computing real radicals and S-radicals of polynomial systems
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An algorithm is proposed, using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety Vℝ(I) as well as a set of generators of theReal radical ideal, obtained in the form of a border or Gröbner basis.
Semidefinite characterization and computation of real radical ideals
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