A Monte Carlo Estimator for an Isolated Polynomial Zero via Contour Integral Representations
Abstract
We introduce a Monte Carlo approach for estimating isolated polynomial zeros through their contour integral representations. For a simple zero enclosed by an isolating contour, we show that the zero can be expressed exactly as the expectation of a complex-valued random variable obtained by uniformly sampling the contour. This yields an unbiased estimator without perturbing the polynomial coefficients. We establish its variance and derive finite sample probabilistic error bounds based on concentration inequalities, with the bounds explicitly reflecting the geometry of the contour and its separation from the zeros. The framework also provides a stochastic formulation of contour based root counting and root isolation. Numerical results illustrate the estimator's behavior under varying polynomial degrees, and we further provide comparisons to existing methods.
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BibTeX
@misc{micheas2026monte,
title = {{A Monte Carlo Estimator for an Isolated Polynomial Zero via Contour Integral Representations}},
author = {Athanasios Christou Micheas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02085},
eprint = {2610.02085},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02085},
}
RIS
TY - UNPB
TI - A Monte Carlo Estimator for an Isolated Polynomial Zero via Contour Integral Representations
AU - Athanasios Christou Micheas
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02085
UR - https://arxiv.org/abs/2610.02085
AB - We introduce a Monte Carlo approach for estimating isolated polynomial zeros through their contour integral representations. For a simple zero enclosed by an isolating contour, we show that the zero can be expressed exactly as the expectation of a complex-valued random variable obtained by uniformly sampling the contour. This yields an unbiased estimator without perturbing the polynomial coefficients. We establish its variance and derive finite sample probabilistic error bounds based on concentration inequalities, with the bounds explicitly reflecting the geometry of the contour and its separation from the zeros. The framework also provides a stochastic formulation of contour based root counting and root isolation. Numerical results illustrate the estimator's behavior under varying polynomial degrees, and we further provide comparisons to existing methods.
ER -