This paper is concerned with certifying that a given point is near an exactroot of an overdetermined or singular polynomial system with rationalcoefficients. The difficulty lies in the fact that consistency ofoverdetermined systems is not a continuous property. Our certification is basedon hybrid symbolic-numeric methods to compute the exact "rational univariaterepresentation" (RUR) of a component of the input system from approximateroots. For overdetermined polynomial systems with simple roots, we compute aninitial RUR from approximate roots. The accuracy of the RUR is increased viaNewton iterations until the exact RUR is found, which we certify using exactarithmetic. Since the RUR is well-constrained, we can use it to certify thegiven approximate roots using alpha-theory. To certify isolated singular roots,we use a determinantal form of the "isosingular deflation", which adds newpolynomials to the original system without introducing new variables. Theresulting polynomial system is overdetermined, but the roots are now simple,thereby reducing the problem to the overdetermined case. We prove that ouralgorithms have complexity that are polynomial in the input plus the outputsize upon successful convergence, and we use worst case upper bounds fortermination when our iteration does not converge to an exact RUR. Examples areincluded to demonstrate the approach.
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