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## Section: New Results

### Global optimization

Let ${f}_{1},\cdots ,{f}_{p}$ be in $ℚ\left[𝐗\right]$, where $𝐗={\left({X}_{1},\cdots ,{X}_{n}\right)}^{t}$, that generate a radical ideal and let $V$ be their complex zero-set. Assume that $V$ is smooth and equidimensional. Given $f\in ℚ\left[X\right]$ bounded below, consider the optimization problem of computing ${f}^{☆}={inf}_{x\in V\cap {ℝ}^{n}}f\left(x\right)$. For $𝐀\in G{L}_{n}\left(ℂ\right)$, we denote by ${f}^{𝐀}$ the polynomial $f\left(𝐀𝐗\right)$ and by ${V}^{𝐀}$ the complex zero-set of ${f}_{1}^{𝐀},...,{f}_{p}^{𝐀}$. In [9] , we construct families of polynomials ${𝖬}_{0}^{𝐀},...,{𝖬}_{d}^{𝐀}$ in $ℚ\left[𝐗\right]$: each ${𝖬}_{i}^{𝐀}$ is related to the section of a linear subspace with the critical locus of a linear projection. We prove that there exists a non-empty Zariski-open set $O\subset G{L}_{n}\left(ℂ\right)$ such that for all $𝐀\in O\cap G{L}_{n}\left(ℚ\right)$, $f\left(x\right)$ is non-negative for all $x\in V\cap {ℝ}^{n}$ if, and only if, ${f}^{𝐀}$ can be expressed as a sum of squares of polynomials on the truncated variety generated by the ideal $〈{𝖬}_{i}^{𝐀}〉$, for $0\le i\le d$. Hence, we can obtain algebraic certificates for lower bounds on ${f}^{☆}$ using semidefinite programs. Some numerical experiments are given. We also discuss how to decrease the number of polynomials in ${𝖬}_{i}^{𝐀}$.