Hereditary NC Polynomials & Positivity Domains
Abstract
Noncommutative polynomials, elements of the free algebra generated by several noncommuting variables x1,...,xg,x1*,...,xg*, are of importance in noncommutative algebra, free analysis and real algebraic geometry. We allow matrix coefficients and view these noncommutative polynomials as functions in several size-free matrix variables. The free locus of a noncommutative polynomial f is the sequence of hypersurfaces 𝒵n(f)={X∈ Mn(ℂ)g:det f(X) = 0}. When f is Hermitian, the positivity domain of f is the sequence of semialgebraic sets 𝒟n(f)={X∈ Mn(ℂ)g:f(X) is positive semidefinite}. We show for well-behaved cases that the Euclidean boundary of the positivity domain is Zariski dense in the free locus. This leads to a certificate for equality of positivity domains of coefficient-irreducible unsignatured atomic quadratics. Finally, we describe the geometric structure of positivity domains of hereditary quadratic polynomials through Γ-convexity.
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BibTeX
@misc{islam2026hereditary,
title = {{Hereditary NC Polynomials & Positivity Domains}},
author = {Abdellah Islam and Jurij Volčič and Hugo J. Woerdeman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14053},
eprint = {2610.14053},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14053},
}
RIS
TY - UNPB
TI - Hereditary NC Polynomials & Positivity Domains
AU - Abdellah Islam
AU - Jurij Volčič
AU - Hugo J. Woerdeman
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.14053
UR - https://arxiv.org/abs/2610.14053
AB - Noncommutative polynomials, elements of the free algebra generated by several noncommuting variables x_1,...,x_g,x_1^*,...,x_g^*, are of importance in noncommutative algebra, free analysis and real algebraic geometry. We allow matrix coefficients and view these noncommutative polynomials as functions in several size-free matrix variables. The free locus of a noncommutative polynomial f is the sequence of hypersurfaces 𝒵_n(f)={X∈ M_n(ℂ)^g:det f(X) = 0}. When f is Hermitian, the positivity domain of f is the sequence of semialgebraic sets 𝒟_n(f)={X∈ M_n(ℂ)^g:f(X) is positive semidefinite}. We show for well-behaved cases that the Euclidean boundary of the positivity domain is Zariski dense in the free locus. This leads to a certificate for equality of positivity domains of coefficient-irreducible unsignatured atomic quadratics. Finally, we describe the geometric structure of positivity domains of hereditary quadratic polynomials through Γ-convexity.
ER -