>From other side the problem to solve systems of equations for the finite parts of the infinite formal series is very important for different applications. For example we could say about investigations of singular points of the systems of ODE. Another example is the problem of normalization of the ODE. [1,2].
Usually one can consider the solutions for the finite parts of the series only. It allow us to consider the system of the equations over the formal power series ring as usuall polynomial system, but this approach does not lead to good result, because adding terms of highest order could change your solutions in lower terms.
We can show that some modification of Buchberger's algotithm allow us to construct a canonical basis whis is full analogue of the GB in the case of ideas in the ring of formal power series. The construction of such canonical basis and the proof of appropriate variant of the Buchberger's theorem about finiteness of the algorithm is very important not only in the connection with the problem to solve the equations in the ring of formal power series but for studing of the structure of the ring of the FPS and it's ideals. This algorithm get us a canonical representation for the ideals of this ring.
We use in our algorithm some special representation for basic elements of the ideal. This representation uses monomials which lie on the border of specially modified Newton polytopes for the series under consideration. For each serie of the system of we have only a finite number of such monomials. If we have the system of finite number of "basic" monomials we can represent the serie as a sum such monomials with coefficients which are also the series. It is very important thing for the algorithm that these coefficients are invertible elements of the ring of FPS. As a result we can evaluate by usual method S polynomials and make reductions of the leading terms only for Newton polytops. It means that each modified Newton polytop of the series of the equations is irreducible relatively others modified Newton Polytops. We suggest to use such sets of FPS as standart basisis in the ring of FPS. Such sets of FPS have many qualities of usual Groebner basisis in the case of polynomials. The proofs of the variants of the Buchberger's theorems are based on the fact that the ring of FPS over a noetherian commutative ring is noetherian also.
References
[1] Bruno A. D. "Local Method in Nonlinear Differential Equations." Part I-The Local Method of Nonlinear Analyses of Differential Equations, Part II-The Sets of Analyticity of a Normalizing Transformation. Springer Series in Soviet Mathematics, 1989. ISBN 3-540-18926-2, 370 p.
[2] Edneral V. F. "Computer Generation of Normalizing Transformation for Systems of Nonlinear ODE." Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation (Kiev, Ukraine, July 6-8, 1993) / M. Bronstein. N.Y.: ACM Press, 1993. P. 14--19.
[3] N.N.Vassiliev, V.F.Edneral, "About an Algorithm to Construct a Canonical Basis for the Ideals in the Ring of Formal Power Series", in the book of Abstracts of the Russian Conference "Computer Methods of Celestial Mechanics -95", St. Petersburg, 1995, Ed. by prof Sokolsky.
