Approximate square-free part and decomposition. (May 2021) - Nagasaka, Kosaku
Abstract: 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.