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Hello,

Could you help me with a reference to elementary properties of Groebner bases in rings of formal power series over a field? I am especially interested in generic initial ideals.

Thank you in advance,
Serge

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    $\begingroup$ Have you looked at Greuel, Pfister, A Singular Introduction to Commutative Algebra? I think it contains a few sections on power series. $\endgroup$ – Michael Bächtold Dec 31 '12 at 9:48
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To expand on Michael's comment, the Greuel, Pfister book Section 6.4 is about standard bases in formal power series rings. Quoting,

The main result is that they can be computed, if the ideal is generated by polynomials. This is the basis for computations in local analytic geometry. The theory of standard bases in power series rings goes back to Hironaka (cf. [123]) and Grauert (cf. [98]).

The references are:

  • Hironaka, H.: Resolution of Singularities of an Algebraic Variety over a Field of Characteristic Zero. Ann. of Math. 79, 109–326 (1964).
  • Grauert, H.: Über die Deformation isolierter Singularitäten analytischer Mengen. Invent. Math. 15, 171–198 (1972).
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The only place where I think I read about Groebner bases for power series was "Coherent Analytic Sheaves" by Grauert and Remmert, but I don't think they got this far.

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There is also a nice treatment of standard bases in the ring of convergent power series in the book of De Jong and Pfister "Local analytic geometry: Basic theory and applications" (it's chapter 7).

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