Approximate square-free part and decomposition

• Published in: Journal of symbolic computation Vol. 104; pp. 402 - 418
• Main Author: Nagasaka, Kosaku
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.05.2021
• Subjects: Approximate polynomial GCD; Symbolic-numeric algorithm for polynomials; Approximate polynomial GCD; Symbolic-numeric algorithm for polynomials
• ISSN: 0747-7171; 1095-855X; 1095-855X
• DOI: 10.1016/j.jsc.2020.08.004

Square-free decomposition is one of fundamental computations for polynomials. However, any conventional algorithm may not work for polynomials with a priori errors on their coefficients. There are mainly two approaches to overcome this empirical situation: approximate polynomial GCD (greatest common divisor) and the nearest singular polynomial. In this paper, we show that these known approaches are not enough for detecting the nearest square-free part (which has no multiple roots) within the given upper bound of perturbations (a priori errors), and we propose a new definition and a new method to detect a square-free part and its decomposition numerically by following a recent framework of approximate polynomial GCD.