Extremal polynomial systems supported on circuits

A real polynomial system with support W ⊂ Zn is called extremal 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 nondegenerate positive solutions of a system supported on a circuit W ⊂ Zn is at most m(W)+1, where m(W) ≤ n is the degeneracy index of W. We prove that if a circuit W ⊂ Zn supports an extremal 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 Zn. In the general case, we prove that any extremal 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 ⊂ Zn which can support extremal 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 extremal systems. Introduction and statement of the main results We consider systems of n polynomial equations in n variables with real coefficients and monomials having integer exponents. The support of such a system is the set of points a ∈ Zn corresponding to monomials xa = x1 1 · · ·xan n appearing with a non-zero coefficient. We are only interested in the solutions in the complex torus (C∗)n and call them simply complex solutions. By Kouchnirenko theorem, the number of isolated complex solutions is bounded from above by the normalized volume of the convex-hull P of W, which is the usual euclidian volume of P scaled by n!. We will always assume that W is not contained in some hyperplane of Rn, for otherwise this volume would vanish. Kouchnirenko’s bound is reached by nondegenerate systems which are systems whose all solutions are non-degenerate, that is, at which the differentials of the defining polynomials are linearly independent. Non-degenerate systems are generic within systems with given support. A solution of a system is called positive if all its coordinates are positive real numbers. A polynomial system is called extremal if all its complex solutions are positive solutions. For simplicity, we consider here only non-degenerate systems whose number of complex solutions is the normalized volume of P (in other words, systems reaching Kouchnirenko’s bound). Any slight perturbation of the coefficient matrix of such a system produces another non-degenerate system with the same support and 2010 Mathematics Subject Classification. 14T05.