Exploiting Chordal Structure in Polynomial Ideals: A Gröbner Bases Approach

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---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 Grobner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Grobner bases algorithms in many cases. The reason is because all computations are done on “smaller” rings of size equal to the treewidth of the graph (instead of the total number of variables). In particular, for a restricted class of i...

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