  Chordal structure and bounded treewidth allow for efficient computation in
numerical linear algebra, graphical models, constraint satisfaction and many
other areas. In this paper, we begin the study of how to exploit chordal
structure in computational algebraic geometry, and in particular, for solving
polynomial systems. The structure of a system of polynomial equations can be
described in terms of a graph. By carefully exploiting the properties of this
graph (in particular, its chordal completions), more efficient algorithms can
be developed. To this end, we develop a new technique, which we refer to as
chordal elimination, that relies on elimination theory and Gr\"obner bases. By
maintaining graph structure throughout the process, chordal elimination can
outperform standard Gr\"obner basis algorithms in many cases. The reason is
that all computations are done on "smaller" rings, of size equal to the
treewidth of the graph. In particular, for a restricted class of ideals, the
computational complexity is linear in the number of variables. Chordal
structure arises in many relevant applications. We demonstrate the suitability
of our methods in examples from graph colorings, cryptography, sensor
localization and differential equations.
