Abstract. We consider the tropical variety T (I) of a prime ideal I generated by the polynomials f 1 , . . . , f r and revisit the regular projection technique introduced by Bieri and Groves from a computational point of view. In particular, we show that I has a short tropical basis of cardinality at most r + codim I + 1 at the price of increased degrees, and we provide a computational description of these bases.

Abstract. We study algebraic and combinatorial aspects of (classical) projections of m-dimensional tropical varieties onto (m + 1)-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work of the authors on projection-based tropical bases, we characterize algebraic properties of the relevant ideals and provide a characterization of the dual subdivision (as a subdivision of a fiber polytope). This dual subdivision naturally leads to the issue of self-intersections of a tropical variety under projections. For the case of curves, we provide some bounds for the (unweighted) number of self-intersections of projections onto the plane and give constructions with many self-intersections.

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We study algebraic and combinatorial aspects of (classical) projections of m-dimensional tropical varieties onto (m + 1)-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work of the authors on projection-based tropical bases, we characterize algebraic properties of the relevant ideals and provide a characterization of the dual subdivision (as a subdivision of a fiber polytope). This dual subdivision naturally leads to the issue of self-intersections of a tropical variety under projections. For the case of curves, we provide some bounds for the (unweighted) number of self-intersections of projections onto the plane and give constructions with many self-intersections. Preliminaries2.1. Tropical geometry. We review some concepts from tropical geometry. As general references see [9,13,15]. Let K be a field with a real valuation as introduced in Section 1 and let T (I) be the tropical variety of an ideal I ✁ K[x 1 , . . . , x n ]. If I is a prime ideal, then by the Bieri-Groves Theorem T (I) is a pure m-dimensional polyhedral complex where m = dim(I) is the Krull dimension of the ideal [2].Define the local cone of a point x of a polyhedral complex ∆ ⊆ R n as the set LC x (∆) := {x + y ∈ R n : ∃ ε > 0 such that {x + ρy : 0 ≤ ρ ≤ ε} ⊆ ∆} .

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