Radoslav Zlatev

Let $X$ be a smooth projective toric variety of dimension~$n-1$ and let $\phi:X\to\mathbb{P}^n$ be a generically finite rational map. The closed image~$Y$ can be defined by a single equation $P(\boldsymbol x)$, called the implicit equation. The implicitization problem asks for techniques for finding the implicit equation.

This is an old problem in algebraic geometry and can be solved effectively through elimination using Gr\"obner bases, for instance. However, this solution represents a black box in terms of geometry and is unfeasible for even rather modest examples.

In this thesis, we develop a framework for both efficiently computing the image~$Y$ of a rational map and studying the relation between algebraic properties of the coordinates $\phi_j$ on one hand, and geometric properties of the image~$Y$, on the other. We follow the theme of moving surfaces by Cox, Goldman and Zhang [2000] and the method of the approximation complex by Bus\'e and Jouanolou [2003] to construct a family of matrices $N$, one for each element in $\text{Pic}(X)$, capturing determinantal representations for the implicit equation.

An algorithm for computing $N$ and $P(\boldsymbol x)$ is described and implemented in the Macaulay2 system. It is benchmarked against standard approaches and later used to carry out computations in previously intractable situations.