Polynomial systems with few real zeroes

Benoît Bertrand (1) , Frederic Bihan (2) , Frank Sottile (3)

1 IUT Tarbes - Institut Mathématique de Toulouse
2 LAMA - Laboratoire de Mathématiques
3 Department of Mathematics [Texas A&M University]

We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds for these systems. When the support is exactly a circuit whose affine span is ${\Z}^n$, this bound is $2n+1$, while the Khovanskii bound is exponential in $n^2$. The bound $2n+1$ can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic.

Domaines: Géométrie algébrique [math.AG]

Benoît Bertrand, Frederic Bihan, Frank Sottile. Polynomial systems with few real zeroes. Mathematische Zeitschrift, 2006, 253, pp.361-385. ⟨hal-00386000⟩