Tropical ideals

Diane Maclagan, University of Warwick. Part of the pure mathematics algebra seminar series.

Tropical geometry studies geometry over the tropical semiring, where multiplication is replaced by addition and addition by minimum. Over the last fifteen years there has been an explosion of work on varieties in this setting. From a commutative algebra perspective, however, the semiring of tropical polynomials is not as nice as the its standard counterpart. We no longer have unique factorization, cancellation, or the Noetherian property.  I will discuss joint work with Felipe Rincon on a special class of "tropical" ideals in this semiring that is much better behaved, and on the geometric side allows us to expand from varieties to schemes. This uses the theory of valuated matroids