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By Garrett P.

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The most function of those lectures is first to in brief survey the elemental con­ nection among the illustration thought of the symmetric workforce Sn and the speculation of symmetric capabilities and moment to teach how combinatorial equipment that come up certainly within the thought of symmetric features result in effective algorithms to specific a variety of prod­ ucts of representations of Sn when it comes to sums of irreducible representations.

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And Sadik used the properties to compute the dimension of J/ = (A): I°° and the differential dimension of a related differential ideal aj = [A]: I°°. Hubert developed similar algorithms using properties of Grobner basis of zero-dimensional polynomial ideals. Due to lack of time and space, we shall not investigate these important results. 11, we saw how a system of arbitrarily high order linear partial differential equations can be simplified, in this case, to a trivial system. This is the essense of elimination theory We are interested in automating this process of simplifying a given system through symbolic computations.

2 We say t h a t A has the Rosenfeld property (resp. strong Rosenfeld property) if every differential polynomial F partially reduced with respect t o A belonging to the differential ideal a # (resp. as) already belongs to t h e ideal JH (resp. Js) in 31. It is easy to see t h a t if A has the strong Rosenfeld property, then it has the Rosenfeld property since the initials I A are partially reduced with respect to A . If A has the Rosenfeld property, then it is possible to answer certain questions about a differential ideal by answering similar questions about an ideal.

Prime) differential ideal p and ifJjj — Ji, then p:H = ajj (resp. p = ajj) a n d the three conditions hold. Proof. 8). 9. For the necessity, suppose A is a characteristic set of a prime differential ideal p. 6), P = <*H (resp. 3). In particular, the Ritt-Kolchin remainder of every A(A,A',v), which belongs to p, is zero. 3, Jj (which is also JH by hypothesis) is radical (resp. 7, JY is radical (resp. prime). Since the initials and separants of A are not in p, they are not in JY. In fact, we may replace V by the subset V consisting of all Oyj G V t h a t appears in some A G A , and JY is radical (resp.

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