In recent years, tropical geometry has developed as a theory on its own. Its two main aims are to answer open questions in algebraic geometry and to give new proofs of celebrated classical results. The main subject of this thesis is concerned with the former: the solution of implicitization problems via tropical geometry. We develop new and explicit techniques that completely solve these challenges in four concrete examples.
We start by studying a family of challenging examples inspired by algebraic statistics and machine learning: the restricted Boltzmann machines F(n,k). These machines are highly structured projective varieties in tensor spaces. They correspond to a statistical model encoded by the complete bipartite graph K(k,n), by marginalizing k of the n+k binary random variables. In Chapter 2, we investigate this problem in the most general setting. We conjecture a formula for the expected dimension of the model, verifying it in all relevant cases. We also study inference functions and their interplay with tropicalization of polynomial maps.
In Chapter 3, we focus on the particular case F(4,2), answering a question by Drton, Sturmfels and Sullivant regarding the degree (and Newton polytope) of the homogeneous equation in 16 variables defining this model. We show that its degree
is 110 and compute its Newton polytope. Along the way, we derive theoretical results in tropical geometry that are crucial for other examples in this thesis, as well as novel computational methods.
In Chapter 4, we study the first secant varieties of monomial projective curves from a tropical perspective. Our main tool is the theory of geometrical tropicalization developed by Hacking, Keel and Tevelev. Their theory hinges on computing the tropicalization of subvarieties of tori by analyzing the combinatorics of their boundary in a suitable (tropical) compactification. We enhance this theory by providing a formula for computing
multiplicities on tropical varieties. We believe that this construction will give insight to understand higher secants of monomial projective curves, which are key objects in toric and birational geometry.
In Chapter 5, we answer the general question of implicitization of parametric surfaces in 3-space via geometric tropicalization. The generic case, together with its higher-dimensional analog, was studied by Sturmfels, Tevelev and Yu. We address this problem for non-generic surfaces. This involves understanding the combinatorics of the intersection of irreducible algebraic curves in the two-dimensional torus and explicitly resolving singularities of points in curves by blow-ups. We conclude with a brief discussion on open problems, including connections to Berkovich spaces, extension of the theory to non-archimedean valued fields and applications of tropical implicitization to classifying tropical surfaces in three-space.