Polynomial systems with few real zeroes Academic Article uri icon

abstract

  • 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 ℤn , 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.

author list (cited authors)

  • Bertrand, B., Bihan, F., & Sottile, F.

citation count

  • 7

publication date

  • February 2006