Root isolation of zero-dimensional polynomial systems with linear univariate representation

• Published in: Journal of symbolic computation Vol. 47; no. 7; pp. 843 - 858
• Main Authors: Cheng, Jin-San, Gao, Xiao-Shan, Guo, Leilei
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.07.2012
• Subjects: Gröbner basis; Linear univariate representation; Local generic position; Root isolation; Zero-dimensional polynomial system; Linear univariate representation; Gröbner basis; Zero-dimensional polynomial system; Root isolation; Local generic position
• ISSN: 0747-7171; 1095-855X
• DOI: 10.1016/j.jsc.2011.12.011

In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the complex roots of the polynomial system are represented as linear combinations of the roots of several univariate polynomial equations. An algorithm is proposed to compute such a representation for a given zero-dimensional polynomial equation system based on Gröbner basis computation. The main advantage of this representation is that the precision of the roots of the system can be easily controlled. In fact, based on the linear univariate representation, we can give the exact precisions needed for isolating the roots of the univariate equations in order to obtain roots of the polynomial system with a given precision. As a consequence, a root isolating algorithm for a zero-dimensional polynomial equation system can be easily derived from its linear univariate representation.