Computing zero-dimensional tropical varieties via projections

Computing Tropical Varieties over Fields with Valuation

We show how tropical varieties of ideals I E K[x] over a field K with non-trivial valuation can always be traced back to tropical varieties of ideals π−1I E RJtK[x] over some dense subring R in its ring of integers. Moreover, for homogeneous ideals, we present algorithms on how the latter can be computed in finite time, provided that π−1I is generated by elements in R[t, x]. While doing so, we ...

Cocycles on Tropical Varieties via Piecewise Polynomials

We use piecewise polynomials to define tropical cocycles generalising the well-known notion of Cartier divisors to higher codimensions. We also introduce an intersection product of cocycles with tropical cycles and prove that this gives rise to a Poincaré duality in some cases.

Equations of Tropical Varieties

We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as T = (R∪{−∞},max,+) by writing them as solution sets to explicit systems of tropical equations that are uniquely determined by tropical linear algebra. We then define a tropicalization func...

On the complexity of computing with zero-dimensional triangular sets

We generalize Kedlaya and Umans’ modular composition algorithm to the multivariate case. As a main application, we give fast algorithms for many operations involving triangular sets (over a finite field), such as modular multiplication, inversion, or change of order. For the first time, we are able to exhibit running times for these operations that are almost linear, without any overhead expone...

Cryptography on Trace-Zero Varieties