Maximally positive polynomial systems supported on circuits

Frédéric Bihan (1)

1 LAMA - Laboratoire de Mathématiques

A real polynomial system with support $\calW \subset \Z^n$ is called {\it maximally positive} if all its complex solutions are positive solutions. A support $\calW$ 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 $\calW \subset\Z^n$ is at most $m(\calW)+1$, where $m(\calW) \leq n$ is the degeneracy index of $\calW$. We prove that if a circuit $\calW \subset \Z^n$ supports a maximally positive system with the maximal number $m(\calW)+1$ 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 $\calW \subset \Z^n$ 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.

Domaines: Géométrie algébrique [math.AG]

Frédéric Bihan. Maximally positive polynomial systems supported on circuits. MEGA'2013 (Special Issue), Jun 2013, Frankfurt am Main, Allemagne. ⟨hal-01022880⟩