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Let $\mathbb{Q}$ (resp. $\mathbb{R}$) be the field of rational (resp. real)numbers and $X = (X_1, \ldots, X_n)$ be variables. Deciding the non-negativityof polynomials in $\mathbb{Q}[X]$ over $\mathbb{R}^n$ or over semi-algebraicdomains defined by polynomial constraints in $\mathbb{Q}[X]$ is a classicalalgorithmic problem for symbolic computation.  The Maple package \textsc{RealCertify} tackles this decision problem bycomputing sum of squares certificates of non-negativity for inputs where suchcertificates hold over the rational numbers. It can be applied to numerousproblems coming from engineering sciences, program verification andcyber-physical systems. It is based on hybrid symbolic-numeric algorithms basedon semi-definite programming.

RealCertify: a Maple package for certifying non-negativity