Maximally positive polynomial systems supported on circuits. (May 2015) - Bihan, Frédéric
Abstract: A real polynomial system with support W ⊂ Z n is called maximally positive if all its complex solutions are positive solutions. A support W having n + 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 ⊂ Z n is at most m ( W ) + 1, where m ( W ) ≤ n is the degeneracy index of W . We prove that if a circuit W ⊂ Z n supports a maximally positive system with the maximal number m ( 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 Z n . 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 ⊂ Z n 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.