Lower Bounds for Real Solutions to Sparse Polynomial Systems

Details Lower Bounds for Real Solutions to Sparse Polynomial Systems Lower Bounds for Real Solutions to Sparse Polynomial Systems

2004-09-27

arxiv.org/abs/math/0409504v2

Advances in Mathematics, Volume 204, Issue 1, 1 August 2006, 116--151.

Mathematics

Algebraic Geometry

31 pages. Minor revisions

Scientific paper

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.

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