41 research outputs found
Lower Bounds for Real Solutions to Sparse Polynomial Systems
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.Comment: 31 pages. Minor revision
Tropical determinant on transportation polytope
Let be the set of all the integer points in the
transportation polytope of matrices with row sums and column
sums . In this paper we find the sharp lower bound on the tropical
determinant over the set . This integer
piecewise-linear programming problem in arbitrary dimension turns out to be
equivalent to an integer non-linear (in fact, quadratic) optimization problem
in dimension two. We also compute the sharp upper bound on a modification of
the tropical determinant, where the maximum over all the transversals in a
matrix is replaced with the minimum.Comment: 16 pages, 2 figure
Toric surface codes and Minkowski length of polygons
In this paper we prove new lower bounds for the minimum distance of a toric
surface code defined by a convex lattice polygon P. The bounds involve a
geometric invariant L(P), called the full Minkowski length of P which can be
easily computed for any given P.Comment: 18 pages, 9 figure