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Excerpt from Symbolic Treatment of Geometric Degeneracies Exact model of numerical computation. It is important to realize that this paper assumes the exact model of numerical computations. All numbers are represented exactly: for example, to represent any real algebraic number or exactly, it suffices to specify a polynomial p(: c) with integer coefficients together with an interval I containing a but no other distinct roots of The end points of I are exact, say represented by rational numbers. Our development is not restricted to any particular exact representation. Note that since the integers involved (as in the coefficients of or in representing I) can be arbitrarily long, we sometimes call this the arbitrary precision model (or, somewhat misleadingly, 'infinite precision This contrasts with the fixed precision world of the numerical analysts. With the advent of computer algebra, the exact models are becoming more important and indeed unavoidable for some applications. We have stated that the issues of fixed precision and data degeneracy are distinct. Indeed, any attempt to consider data degeneracy in the fixed precision models faces some very difficult problems: for instance, there are many puzzles in just trying to define what it means for three points to be collinear in the fixed precision model (actually, the world of pixels). In any case, one should begin by understanding degeneracy in exact models. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

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