A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems

Xiaojie Dou and Jin-San Cheng
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Xiaojie Dou: College of Science, Civil Aviation University of China, Tianjin 300300, China
Jin-San Cheng: KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Mathematics, 2018, vol. 6, issue 9, 1-18

Abstract: In this paper, by transforming the given over-determined system into a square system, we prove a necessary and sufficient condition to certify the simple real zeros of the over-determined system by certifying the simple real zeros of the square system. After certifying a simple real zero of the related square system with the interval methods, we assert that the certified zero is a local minimum of sum of squares of the input polynomials. If the value of sum of squares of the input polynomials at the certified zero is equal to zero, it is a zero of the input system. As an application, we also consider the heuristic verification of isolated zeros of polynomial systems and their multiplicity structures.

Keywords: over-determined polynomial system; isolated zeros; minimum point; sum of squares; interval methods (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
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