Lower Bounds for Real Solutions to Sparse Polynomial Systems
Algebraic Geometry
2010-03-29 v2 Combinatorics
Abstract
We show how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the sign-imbalance of P and it holds if all maximal chains of P have length of the same parity. This theory also gives lower bounds in the real Schubert calculus through sagbi degeneration of the Grassmannian to a toric variety, and thus recovers a result of Eremenko and Gabrielov.
Cite
@article{arxiv.math/0409504,
title = {Lower Bounds for Real Solutions to Sparse Polynomial Systems},
author = {Evgenia Soprunova and Frank Sottile},
journal= {arXiv preprint arXiv:math/0409504},
year = {2010}
}
Comments
31 pages. Minor revisions