Tree-width in Algebraic Complexity
The paper surveys some of the author's work studying the algorithmic importance of the tree-width notion in algebraic frameworks. Two approaches are described. The first gives an algorithmic meta-theorem for certain logically characterized properties within the Blum-Shub-Smale BSS model of computation over the reals. The second reports on recent joint work with P. Koiran relating Boolean complexity and Valiant's approach to study families of polynomial systems over infinite fields and their complexity. We define particular families of polynomials via bounding the tree-width of suitably attached graphs and study the expressive power of the resulting families. The work described here is partially co-authored with and partially very much influenced by previous work of Janos A. Makowsky.
tree-width real number computations descriptive complexity valiant's model