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.
Highlights
Finding zeros of polynomial systems is a fundamental problem in scientific computing.Newtonâs method is widely used to solve this problem
We prove that the simple real zeros of the input system are local minima of sum of squares of the input polynomials
From Theorem 1, we know that the simple real zeros of ÎŁ and ÎŁr are in one-to-one correspondence with the constraint that the value of the sum of squares of the polynomials in ÎŁ at the simple real zeros is identically zero
Summary
Finding zeros of polynomial systems is a fundamental problem in scientific computing. We consider the problem of certifying the simple real zeros of an over-determined polynomial system. After transforming the i =1 input over-determined system into a square one, we can use both the α-theory and the interval methods to certify its simple zeros. We only consider using the interval methods to certify the simple real zeros of the over-determined system. We get a necessary and sufficient condition to certify the simple real zeros of the input system Σ by certifying the simple real zeros of the square system Σr. A big difference between this paper and our pervious work [32] is that we do not merely consider certifying simple zeros of over-determined polynomial systems, and consider the certification of the general isolated zeros.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
