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            <title xml:lang="en">Deciding non-negativity of polynomials based on Sturm's theorem and Descartes' rule of signs</title>
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                <forename type="first">Nguyen Hong</forename>
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              <p>&lt;div&gt;&lt;p&gt;We consider the problem of deciding the non-negativity and positivity of an $n$-variate polynomial $p$ over a real algebraic set $S$ defined by  polynomials $g=(g_1,\dots,g_m)$. In order to accomplish this objective, we employ the critical value polynomial $\varphi_{p,g}$ whose roots are the complex critical values of $p$  over the complex algebraic set defined by $g$. Firstly, we demonstrate that, under certain genericity conditions, the decision problem is equivalent to determining negative or non-positive real roots of $\varphi_{p,g}$. Therefore, Sturm's theorem and Descartes' rule of signs are used to solve the problem in the general and convex settings, respectively.Secondly, we propose symbolic algorithms that can decide the non-negativity or positivity of $p$ over $S$, and estimate their bit complexities in scenarios where the input data is characterized by rational coefficients. Finally, we apply these results to decide the convexity or strict convexity of $p$ over the ambient space $\mathbb{R}^n$.&lt;/p&gt;&lt;/div&gt;</p>
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