By Evelyne Tournier

The pc Algebra and Differential Equations assembly held in France in June 1992 (CADE-92) was once the 3rd of a sequence of biennial workshops dedicated to fresh advancements in machine algebra structures. This booklet includes chosen papers from that assembly. 3 major subject matters are mentioned. the 1st of those is the speculation of D-modules. this provides a good technique to successfully deal with linear platforms of partial differential equations. the second one subject issues the theoretical elements of dynamical structures, with an advent to Ecalle idea and perturbation research utilized to differential equations and different nonlinear structures. the ultimate subject is the speculation of standard kinds. right here contemporary advancements within the thought and computation of standard types are mentioned.

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**Extra resources for Computer Algebra and Differential Equations**

**Sample text**

By definition b/(s) — bj(s)/{s + I) is the minimal polynomial of the action of s on (s + \){V[s]f/V[s\f'+1). 6 (s + l)(V[s)f'/V[s}r+1) ~ V[8]f'/V[s]{f, J(f))f , Proof : The only difficulty is to prove that this application is well defined. Let P € T>[s] such that (s + \)Pf = Q(s)f'+1 where Q(s) € V[s\. We write Q{s) = (s + l)T(s) + R where T{s) e V[a) , R€V . Take s = - 1 , we find R(l) = 0. Then we can write R = YlRidi- And we obtain Hence, the application is well defined. 8 (B. Malgrange) b/(s) is the minimal polynomial of the action of s on We now take /i elements C independent of O {e\,...

48 (1984) . [S] C. , Hermann, to appear. [W] W. , Interscience, Publ. (1965).

And c(IT(V[s]f')) of YA=I E& c a n be achieved after a finite number of computations. With standard basis technique, all computations are algorithmic. Thus, we have produced an algorithm to compute the Bernstein polynomial for an isolated singularity. 10 Let c(s) = bf(s)... bf(s + \i - 1) and c(s) = (s 4- l)c(s) 1. We have the following precise functional equations: 3AeT> (independent of s) c(s)f = Afs+1 such that . and degree(A) < deg(c(s)) = /x(/x - 1 ) + 1 , 2. We have the following precise operator in the annihilator of f: with B eT> independent of s and degree (B) < deg(c(s)) = fj,(n - 1) .