Every tropical ideal in the sense of Maclagan-Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements, and in which all maximal cones have weight one.

Abstract. Let f be a generically finite polynomial map f : C n → C m of algebraic degree d. Motivated by the study of the Jacobian Conjecture, we prove that the set S f of nonproperness of f is covered by parametric curves of degree at most d − 1. This bound is best possible. Moreover, we prove that if X ⊂ R n is a closed algebraic set covered by parametric curves, and f : X → R m is a generically finite polynomial map, then the set S f of non-properness of f is also covered by parametric curves. Moreover, if X is covered by parametric curves of degree at most d 1 , and the map f has degree d 2 , then the set S f is covered by parametric curves of degree at most 2d 1 d 2 . As an application of this result we show a real version of the Białynicki-Birula theorem: Let G be a real, non-trivial, connected, unipotent group which acts effectively and polynomially on a connected smooth algebraic variety X ⊂ R n . Then the set Fi x(G) of fixed points has no isolated points.