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WINTER 2015 Monomial valuations, cusp singularities, and continued fractions
David J. Bruce, Molly Logue, Robert Walker
J. Commut. Algebra 7(4): 495-522 (WINTER 2015). DOI: 10.1216/JCA-2015-7-4-495


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 $a/b$.


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David J. Bruce. Molly Logue. Robert Walker. "Monomial valuations, cusp singularities, and continued fractions." J. Commut. Algebra 7 (4) 495 - 522, WINTER 2015.


Published: WINTER 2015
First available in Project Euclid: 19 January 2016

zbMATH: 1329.13037
MathSciNet: MR3451353
Digital Object Identifier: 10.1216/JCA-2015-7-4-495

Primary: 13F30, 14E15, 16W60

Rights: Copyright © 2015 Rocky Mountain Mathematics Consortium


Vol.7 • No. 4 • WINTER 2015
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