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Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. However, it can be expensive, with worst case complexity doubly exponential in the size of the input. Hence it is important to formulate the problem in the best manner for the CAD algorithm. One possibility is to precondition the input polynomials...
In considering the reliability of numerical programs, it is normal to ``limit our study to the semantics dealing with numerical precision'' (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that essentially ignores the numerics. The thesis of this paper is that there is a class of problems that fall between these two, which could be described as ``does...
In computer algebra there are different ways of approaching the mathematical concept of functions, one of which is by defining them as solutions of differential equations. We compare different such approaches and discuss the occurring problems. The main focus is on the question of determining possible branch cuts. We explore the extent to which the treatment of branch cuts can be rendered (more) algorithmic,...
While much is written about the importance of sparse polynomials in computer algebra, much less is known about the complexity of advanced (i.e. anything more than multiplication!) algorithms for them. This is due to a variety of factors, not least the problems posed by cyclotomic polynomials. In this paper we state a few of the challenges that sparse polynomials pose.
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