A bound on the minimum of a real positive polynomial over the standard simplex.
Résumé
We consider the problem of bounding away from $0$ the minimum value $m$ taken by a polynomial $P \in \Z \left[X_{1},\dots,X_{k}\right]$ over the standard simplex $\Delta \subset \R^{k}$, assuming that $m>0$. Recent algorithmic developments in real algebraic geometry enable us to obtain a positive lower bound on $m$ in terms of the dimension $k$, the degree $d$ and the bitsize $\tau$ of the coefficients of $P$. The bound is explicit, and obtained without any extra assumption on $P$, in contrast with previous results reported in the literature.
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