## Section: New Results

### Global optimization

Let ${f}_{1},\cdots ,{f}_{p}$ be in $\mathbb{Q}\left[\mathbf{X}\right]$, where $\mathbf{X}={({X}_{1},\cdots ,{X}_{n})}^{t}$, that generate a radical ideal and let $V$ be their complex zero-set. Assume that $V$ is smooth and equidimensional. Given $f\in \mathbb{Q}\left[X\right]$ bounded below, consider the optimization problem of computing ${f}^{\u2606}={inf}_{x\in V\cap {\mathbb{R}}^{n}}f\left(x\right)$. For $\mathbf{A}\in G{L}_{n}\left(\u2102\right)$, we denote by ${f}^{\mathbf{A}}$ the polynomial $f\left(\mathbf{A}\mathbf{X}\right)$ and by ${V}^{\mathbf{A}}$ the complex zero-set of ${f}_{1}^{\mathbf{A}},...,{f}_{p}^{\mathbf{A}}$. In [9] , we construct families of polynomials ${\U0001d5ac}_{0}^{\mathbf{A}},...,{\U0001d5ac}_{d}^{\mathbf{A}}$ in $\mathbb{Q}\left[\mathbf{X}\right]$: each ${\U0001d5ac}_{i}^{\mathbf{A}}$ 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 $O\subset G{L}_{n}\left(\u2102\right)$ such that for all $\mathbf{A}\in O\cap G{L}_{n}\left(\mathbb{Q}\right)$, $f\left(x\right)$ is non-negative for all $x\in V\cap {\mathbb{R}}^{n}$ if, and only if, ${f}^{\mathbf{A}}$ can be expressed as a sum of squares of polynomials on the truncated variety generated by the ideal $\langle {\U0001d5ac}_{i}^{\mathbf{A}}\rangle $, for $0\le i\le d$. Hence, we can obtain algebraic certificates for lower bounds on ${f}^{\u2606}$ using semidefinite programs. Some numerical experiments are given. We also discuss how to decrease the number of polynomials in ${\U0001d5ac}_{i}^{\mathbf{A}}$.