HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information

# Punctual Hilbert Schemes and Certified Approximate Singularities

1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , NKUA - National and Kapodistrian University of Athens
Abstract : In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f $=(f_1, \ldots, f_N)\in C[x_1, \ldots, x_n]^N$, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of $f$ such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so called inverse system that describes the multiplicity structure at the root. We use $α$-theory test to certify the quadratic convergence, and togive bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.
Keywords :
Document type :
Conference papers
Domain :
Complete list of metadata

Cited literature [38 references]

https://hal.inria.fr/hal-02478768
Contributor : Bernard Mourrain Connect in order to contact the contributor
Submitted on : Tuesday, June 30, 2020 - 10:31:05 AM
Last modification on : Friday, February 4, 2022 - 3:33:43 AM

### Files

paper.pdf
Files produced by the author(s)

### Citation

Angelos Mantzaflaris, Bernard Mourrain, Agnes Szanto. Punctual Hilbert Schemes and Certified Approximate Singularities. ISSAC 2020 - International Symposium on Symbolic and Algebraic Computation, Jul 2020, Kalamata, Greece. ⟨10.1145/3373207.3404024⟩. ⟨hal-02478768v2⟩

Record views