Symbolic Computation with Symmetric Polynomials an Extension to Macsyma

Abstract : This paper presents SYM, an extension package to MACSYMA symbolic manipulation system. SYM allows to deal efficiently with symetric polynomials which appear to be useful in several symbolic algorithms such as the computation of resolvants, resultants or minimal polynomials. The SYM package proposes various algorithms which allows to test the symetricity of polynomials, to compute orbits of symetric polynomials, to make algebraic operations on them, to realize change of basis for the symetric polynomials algebra and so on. Our algorithms are very efficient because they use a particular representation of the symetric polynomials using only one monomial associated with each orbit. This contracted representation has been possible thanks to the symetricity of the polynomials that we consider. The main resulting property is that we avoid the combinatory explosion associated to the exponential development of the symetric group of the variables of a polynomial. Like MACSYMA, the SYM extension is written in the Lisp programming language because it uses heavily MACSYMA primitive tools. SYM has been developped under UNIX 4.3 bsd in Franzlisp. Our presentation will include a description of the SYM package and some interesting applications built on SYM. The paper will be an english version of the joined presentation of the system. INTRODUCTION We present here a package of manipulations of symmetric polynomials implemented in Fran-zlisp. This package, called SYM, constitutes at present an extension of the system of symbolic computation MACSYMA. It performs a few manipulations on symmetric polynomials; it can also be used for direct applications. Some algorithms extend easily to functions that are symmetric with respect to sets of variables (i.e. multi-symmetric functions); these functions will be dealt with in the present paper.
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Annick Valibouze. Symbolic Computation with Symmetric Polynomials an Extension to Macsyma. Erich Kaltofen and Stephen M. Watt. Computers and Mathematics , ⟨Springer-Verlag NY⟩, pp.308-320, 1989, 978-0-387-97019-6
 ⟨10.1007/978-1-4613-9647-5_35⟩. ⟨hal-01672106⟩

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