Dou , X.; Cheng , J.-S. A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems. Mathematics2018, 6, 166.
Dou , X.; Cheng , J.-S. A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems. Mathematics 2018, 6, 166.
We construct a real square system related to a given over-determined real system. We prove that the simple real zeros of the over-determined system are the simple real zeros of the related square system and the real zeros of the two systems are one-to-one correspondence with the constraint that the value of the sum of squares of the polynomials in the over-determined system at the real zeros is identically zero. After certifying the simple real zeros of the related square system with the interval methods, we assert that the certified zero is a local minimum of the sum of squares of the input polynomials. If the value of the sum of the 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 the isolated zeros of polynomial systems and their multiplicity structures. Notice that a complex system with complex zeros can be transformed into a real system with real zeros.
over-determined polynomial system; isolated zeros; minimum point; sum of squares; interval methods
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