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- Alternative Ways of Solving Polynomial Systems, August 24, 2001, Ilias Kotsireas and Gregory Reid, 4 pages
**Abstract**We exploit an isomorphic embedding of polynomial systems in linear systems of PDE, to solve polynomials using differential techniques. In theory such methods should already be present in the polynomial sector of the computer algebra system, and if present should be more efficient, since the overhead of polynomial operations should be lower than that for differential systems. In practice however, the varying amounts of programming effort in different sectors of a computer algebra systems will always mean that some methods will be present in some sectors and not in others. Sometimes this occurs in surprising ways, as in this note. We suggest that constructing conversion tools between different sectors of a computer algebra system is a valuable activity. We illustrate such tools in this paper to convert polynomial systems to differential systems. Some problems are treated which cannot be solved using the existing polynomial algorithms implemented in the computer algebra system we used (Maple 7). We present some ways to solve polynomial systems using the differential elimination algorithm rifsimp and the notion of normal set which arises in the interpretation of polynomial system solving as a matrix eigenproblem. These ideas apply to zero-dimensional systems, that is to systems with finitely many solutions. For such systems we show how to compute the associated univariate polynomials satisfied by a single variable of the system. We also treat some systems with parameters, for which we identify several cases in parameter space with their associated solutions. These ideas are illustrated with two zero-dimensional polynomial systems that arise in the study of central configurations in the N-body problem of Celestial Mechanics and an inverse kinematics exmaple from Robotics. The computations have been performed in Maple 6.

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