Algebraic Matroids and Set-theoretic Realizability of Tropical Varieties

摘要

To each prime ideal in a polynomial ring over a field we associate analgebraic matroid and show that it is preserved under tropicalization. Thisgives a necessary condition for a tropical variety to be set-theoreticallyrealizable from a prime ideal. We also show that there are infinitely manyBergman fans that are not set-theoretically realizable as the tropicalizationof any ideal.