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The conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications. We demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres. We survey recent developments and illustrate our approach by means of examples.

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Abstract:We consider the influence of higher-order terms in the leptodermous expansion used to extract the incompressibility K, of infinite nuclear matter from data on the breathing mode of finite nuclei. The terms we calculate are the curvature term Z&A -2'3, the surface-symmetry term Z& Z2A-"3, the quartic volume-symmetry term Z&Z+', and a Coulomb-exchange term. Working within the framework of the scaling model we derive expressions for their coefficients in terms of quantities that are defined for infinite and semi-infinite nuclear matter. We calculate these coefficients for four different Skyrme-type forces, using the extended Thomas-Fermi (ETF) approximation. With the same forces we also calculate the incompressibility K(A, I) for a number of finite nuclei, fit the results to the leptodetmous expansion, and thereby extract new results for the same coefficients. The comparison of the two calculations shows that the leptodermous expansion is converging rapidly. Of the new terms, the term K,Z4 is quite negligible, the curvature term should be included, and we discuss to what extent the other higher-order terms are significant.