Characterization of circuits supporting polynomial systems with the maximal number of positive solutions

Abstract: A polynomial system with $n$ equations in $n$ variables supported on a set $\mathcal{W}\subset\mathbb{R}^n$ of $n+2$ points has at most $n+1$ non-degenerate positive solutions. Moreover, if this bound is reached, then $\mathcal{W}$ is minimally affinely dependent, in other words, it is a circuit in $\mathbb{R}^n$. For any positive integer number $n$, we determine all circuits $\mathcal{W}\subset\mathbb{R}^n$ which can support a polynomial system with $n+1$ non-degenerate positive solutions. Restrictions on such circuits $\mathcal{W}$ are obtained using Grothendieck's real dessins d'enfant, while polynomial systems with $n+1$ non-degenerate positive solutions are constructed using Viro's combinatorial patchworking.