The "Virtual Paper" section provides hyperlinks to already published papers in other journals, which are selected by the editors as excellent papers in topological science and technology.

The tropical variety of a d-dimensional prime ideal in a polynomial ring with complex coefficients is a pure d-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Gröbner fan software Gfan. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms.

Tropicalization is a procedure that takes subvarieties of an algebraic torus to balanced weighted rational complexes in space. In this paper, we study the tropicalizations of curves in surfaces in 3-space. These are balanced rational weighted graphs in tropical surfaces. Specifically, we study the 'lifting' problem: given a graph in a tropical surface, can one find a corresponding algebraic curve in a surface? We develop specific combinatorial obstructions to lifting a graph by reducing the problem to the question of whether or not one can factor a polynomial with particular support in the characteristic 0 case. This explains why some unusual tropical curves constructed by Vigeland are not liftable.

Parametric Presburger arithmetic concerns families of sets S t ⊆ Z d , for t ∈ N, that are defined using addition, inequalities, constants in Z, Boolean operations, multiplication by t, and quantifiers on variables ranging over Z. That is, such families are defined using quantifiers and Boolean combinations of formulas of the formRecent results of Chen, Li, Sam; Calegari, Walker; Roune, Woods; and Shen concern specific families in parametric Presburger arithmetic that exhibit quasi-polynomial behavior. For example, |S t | might be a quasi-polynomial function of t or an element x(t) ∈ S t might be specifiable as a function with quasi-polynomial coordinates, for sufficiently large t. Woods conjectured that all parametric Presburger sets exhibit this quasi-polynomial behavior. Here, we prove this conjecture, using various tools from logic and combinatorics.

We prove that the determinantal complexity of a hypersurface of degree d > 2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the 3 × 3 permanent is 7. We also prove that for n > 3, there is no nonsingular hypersurface in P n of degree d that has an expression as a determinant of a d × d matrix of linear forms, while on the other hand for n ≤ 3, a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5.