The Moment Method for Computing Real Points of Sparse Polynomial Systems
Abstract
One of the most important problems in computational algebra is the computation of real solutions of a polynomial system. Lasserre, Laurent, and Rostalski introduced a numerical method to compute the real radical of an ideal defining a finite set of real points. Their method is based on moment matrices and reduces the problem to solving a semidefinite program. In this work, we build on this method and on the work of Laurent and Mourrain to exploit the sparsity of the input polynomials. For this purpose, we extend the theory of moment matrices to the setting of affine toric varieties. We prove that, for sparse polynomial systems, this approach allows us to recover real points on affine toric varieties. Moreover, we show that a projection of the spectrahedron associated with the semidefinite program is, in fact, a polytope, and we propose a variation of the method to compute only one real solution of the system.
Related
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- Tempered Langevin diffusions with polynomial-tailed targets: explosions, moment thresholds and ergodicity
- The discrete second moment of the Riemann zeta-function along powers of Gram points
- Convergence rate of the moment-SOS hierarchy for univariate polynomial optimization
BibTeX
@misc{baumbach2026moment,
title = {{The Moment Method for Computing Real Points of Sparse Polynomial Systems}},
author = {Tom Baumbach and Matías Bender},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33313},
eprint = {2609.33313},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33313},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Moment Method for Computing Real Points of Sparse Polynomial Systems
AU - Tom Baumbach
AU - Matías Bender
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33313
UR - https://arxiv.org/abs/2609.33313
AB - One of the most important problems in computational algebra is the computation of real solutions of a polynomial system. Lasserre, Laurent, and Rostalski introduced a numerical method to compute the real radical of an ideal defining a finite set of real points. Their method is based on moment matrices and reduces the problem to solving a semidefinite program. In this work, we build on this method and on the work of Laurent and Mourrain to exploit the sparsity of the input polynomials. For this purpose, we extend the theory of moment matrices to the setting of affine toric varieties. We prove that, for sparse polynomial systems, this approach allows us to recover real points on affine toric varieties. Moreover, we show that a projection of the spectrahedron associated with the semidefinite program is, in fact, a polytope, and we propose a variation of the method to compute only one real solution of the system.
ER -