Journal of Commutative Algebra

Monomial valuations, cusp singularities, and continued fractions

David J. Bruce, Molly Logue, and Robert Walker

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Abstract

This paper explores the relationship between real valued monomial valuations on $k(x,y)$, the resolution of cusp singularities and continued fractions. It is shown that, up to equivalence, there is a one-to-one correspondence between real valued monomial valuations on $k(x,y)$ and continued fraction expansions of real numbers between zero and one. This relationship with continued fractions is then used to provide a characterization of the valuation rings for real valued monomial valuations on $k(x,y)$. In the case when the monomial valuation is equivalent to an integral monomial valuation, we exhibit explicit generators of the valuation rings. Finally, we demonstrate that, if $\nu$ is a monomial valuation such that $\nu(x)=a$ and $\nu(y)=b$, where $a$ and $b$ are relatively prime positive integers larger than one, then $\nu$ governs a resolution of the singularities of the plane curve $x^{b}=y^{a}$ in a way we make explicit. Further, we provide an exact bound on the number of blow ups needed to resolve singularities in terms of the continued fraction of $a/b$.

Article information

Source
J. Commut. Algebra, Volume 7, Number 4 (2015), 495-522.

Dates
First available in Project Euclid: 19 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.jca/1453211671

Digital Object Identifier
doi:10.1216/JCA-2015-7-4-495

Mathematical Reviews number (MathSciNet)
MR3451353

Zentralblatt MATH identifier
1329.13037

Subjects
Primary: 13F30: Valuation rings [See also 13A18] 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 16W60: Valuations, completions, formal power series and related constructions [See also 13Jxx]

Keywords
Monomial valuations resolution of singularities continued fractions

Citation

Bruce, David J.; Logue, Molly; Walker, Robert. Monomial valuations, cusp singularities, and continued fractions. J. Commut. Algebra 7 (2015), no. 4, 495--522. doi:10.1216/JCA-2015-7-4-495. https://projecteuclid.org/euclid.jca/1453211671


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