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Tree Breadth of the Continued Fractions Root Finding Method

The continued fractions algorithm for isolating the real roots of polynomials with certainty is one of the most efficient known and widely used. It can be viewed as exploring a tree created by a sequence of simple transformations. In this paper we produce new upper bounds for the breadth of the tree that are significantly smaller than the degree of the input polynomial. We also consider the expected breadth under a reasonable distribution and derive a bound, subject to a plausible assumption, that grows logarithmically with the degree and coefficient size.