Tristram Bogart

Computing tropical varieties

The tropical variety of a d-dimensional prime ideal in a polynomial ring with complex coefficients is a pure d-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Grobner fan software Gfan. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms.

A Lower Bound for the Determinantal Complexity of a Hypersurface

We prove that the determinantal complexity of a hypersurface of degree

Finitely many smooth d-polytopes with N lattice points

d > 2

Superlinear subset partition graphs with dimension reduction, strong adjacency, and endpoint count

d>2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the

Riffs on the infinite ping-pong ball conundrum

3 \times 3

Obstructions to Lifting Tropical Curves in Surfaces in 3-Space

3×3 permanent is 7. We also prove that for