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Constructive invariant theory for tori

David Wehlau (1993)

Annales de l'institut Fourier

Consider a rational representation of an algebraic torus T on a vector space V . Suppose that { f 1 , , f p } is a homogeneous minimal generating set for the ring of invariants, k [ V ] T . New upper bounds are derived for the number N V , T : = max { deg f i } . These bounds are expressed in terms of the volume of the convex hull of the weights of V and other geometric data. Also an algorithm is described for constructing an (essentially unique) partial set of generators { f 1 , , f s } consisting of monomials and such that k [ V ] T is integral over k [ f 1 , , f s ] .

Continuous division of differential operators

Herwig Hauser, Luis Narváez-Macarro (2001)

Annales de l’institut Fourier

In this paper, we give a new proof of the continuity of division by differential operators with analytic coefficients, originally proved by Mebkhout and the second author. Our methods come from the proof of the Constant Rank Theorem for analytic maps between power series spaces, given by Müller and the first author.

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