Fast Computation of the Kippenhahn Hypersurface of a Joint Numerical Range
Abstract
We present a fast modular interpolation method for computing the hypersurface components of algebraic Kippenhahn varieties over number fields. Starting from Hermitian matrices, we factor their determinantal polynomial and use Katz's test to identify the factors whose duals are hypersurfaces. For each such factor, we sample smooth points over finite fields and compute their tangent hyperplanes. These give points on the dual hypersurface, from which we interpolate its defining equation. For number-field coefficients, we use primes at which the defining polynomial of the field factors completely into distinct linear factors. Interpolation across all its roots recovers the residues of the coefficients in a fixed number-field basis. Chinese remaindering and rational reconstruction then recover an equation over the original number field from the computations at different primes. An exact polynomial divisibility test and an irreducibility check certify each reconstructed equation. Four determinantal quartics arising from matrix examples in the literature yield certified dual surfaces of degrees $12$, $18$, $24$ and $34$. We give implementations in Magma and Singular and compare their running times with Gröbner basis elimination.
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BibTeX
@misc{rito2026fast,
title = {{Fast Computation of the Kippenhahn Hypersurface of a Joint Numerical Range}},
author = {Carlos Rito},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.14543},
eprint = {2611.14543},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.14543},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fast Computation of the Kippenhahn Hypersurface of a Joint Numerical Range
AU - Carlos Rito
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.14543
UR - https://arxiv.org/abs/2611.14543
AB - We present a fast modular interpolation method for computing the hypersurface components of algebraic Kippenhahn varieties over number fields. Starting from Hermitian matrices, we factor their determinantal polynomial and use Katz's test to identify the factors whose duals are hypersurfaces. For each such factor, we sample smooth points over finite fields and compute their tangent hyperplanes. These give points on the dual hypersurface, from which we interpolate its defining equation. For number-field coefficients, we use primes at which the defining polynomial of the field factors completely into distinct linear factors. Interpolation across all its roots recovers the residues of the coefficients in a fixed number-field basis. Chinese remaindering and rational reconstruction then recover an equation over the original number field from the computations at different primes. An exact polynomial divisibility test and an irreducibility check certify each reconstructed equation. Four determinantal quartics arising from matrix examples in the literature yield certified dual surfaces of degrees $12$, $18$, $24$ and $34$. We give implementations in Magma and Singular and compare their running times with Gröbner basis elimination.
ER -