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arXiv · 1305.0775

Residual ideals of MacLane valuations

Julio Fernández·Jordi Guàrdia·Jesús Montes·Enric Nart
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Abstract

Let $K$ be a field equipped with a discrete valuation $v$. In a pioneering work, S. MacLane determined all valuations on $K(x)$ extending $v$. His work was recently reviewed and generalized by M. Vaquié, by using the graded algebra of a valuation. We extend Vaquié's approach by studying residual ideals of the graded algebra of a valuation as an abstract counterpart of certain residual polynomials which play a key role in the computational applications of the theory. As a consequence, we determine the structure of the graded algebra of the discrete valuations on $K(x)$ and we show how these valuations may be used to parameterize irreducible polynomials over local fields up to Okutsu equivalence.

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Julio Fernández, Jordi Guàrdia, Jesús Montes, Enric Nart. 2013-07-26. Residual ideals of MacLane valuations. https://arxiv.org/abs/1305.0775

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