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Polynomial systems can be solved reliably using numerical homotopy methods. These methods return numerical approximations to solutions, and all the implementations validate the solutions heuristically. Therefore, the output, the approximate solutions of polynomial systems are not certified. Even though the approximate solutions work well in practice, they cannot be used in critical applications, especially in pure mathematics or when high precision is needed (eg. Surgical Robot arm applications).

Let $I$ be a zero dimensional and radical ideal generated by $m$ polynomials with exact rational coefficients. Assume that we are given approximations for the common exact roots. 

In this talk, we show how to construct and certify the rational entries of Hermite matrices for $I$ from the approximate roots. Furthermore, we represent a method to certify the real roots of the given polynomial system using the signature of Hermite matrices.