Maximally positive polynomial systems supported on circuits

A real polynomial system with support W⊂ZnW⊂Zn is called maximally positive   if all its complex solutions are positive solutions. A support WW having n+2n+2 elements is called a circuit. We previously showed that the number of non-degenerate positive solutions of a system supported on a circuit W⊂ZnW⊂Zn is at most m(W)+1m(W)+1, where m(W)≤nm(W)≤n is the degeneracy index of WW. We prove that if a circuit W⊂ZnW⊂Zn supports a maximally positive system with the maximal number m(W)+1m(W)+1 of non-degenerate positive solutions, then it is unique up to the obvious action of the group of invertible integer affine transformations of ZnZn. In the general case, we prove that any maximally positive system supported on a circuit can be obtained from another one having the maximal number of positive solutions by means of some elementary transformations. As a consequence, we get for each n   and up to the above action a finite list of circuits W⊂ZnW⊂Zn which can support maximally positive polynomial systems. We observe that the coefficients of the primitive affine relation of such circuit have absolute value 1 or 2 and make a conjecture in the general case for supports of maximally positive systems.