Abstract
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.
Highlights
An ideal in a polynomial ring over a field with a non-Archimedean valuation gives rise to a tropical variety, either by taking all weight vectors whose initial ideals do not contain a monomial or, equivalently if the field and the value group are large enough [4, Theorem 4.2], by applying the coordinate-wise valuation to all points in the zero set of the ideal
5 Not every Bergman fan is the variety of a tropical ideal We prove that not every balanced polyhedral complex can be obtained as the variety of a tropical ideal
6 Concluding remarks Using the result by Las Vergnas that U2,3 and V8 do not have a quasi-product of rank 8, we have showed that the Bergman fan of their direct sum is not the tropical variety of any tropical ideal, with weight 1 on all the maximal cones
Summary
An ideal in a polynomial ring over a field with a non-Archimedean valuation gives rise to a tropical variety, either by taking all weight vectors whose initial ideals do not contain a monomial or, equivalently if the field and the value group are large enough [4, Theorem 4.2], by applying the coordinate-wise valuation to all points in the zero set of the ideal. In the middle of this construction sits a tropical ideal, obtained by applying the valuation to all polynomials in the ideal This ideal is a purely tropical object, in that it does not know about the field or the valuation, and it contains more information than the tropical variety itself. It was proved in [6] that tropical ideals, while not finitely generated as ideals—nor in any sense that we know of!—have a rational Hilbert series, satisfy the ascending chain condition, and define a tropical variety: a finite weighted polyhedral complex. Later in [7], it was shown that the top-dimensional parts of these varieties are always balanced polyhedral complexes This leads to the following realisability question. Question 1.1 Which pure-dimensional balanced polyhedral complexes are the variety of some tropical ideal?.
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