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**Harvard**

Sznajdman, J. (2010) *Some Analytic generalizations of the Briancon-Skoda Theorem*. Göteborg : Chalmers University of Technology and University of Gothenburg (Preprint - Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University, nr: ).

** BibTeX **

@book{

Sznajdman2010,

author={Sznajdman, Jacob},

title={Some Analytic generalizations of the Briancon-Skoda Theorem},

abstract={<p>The Brian\c con-Skoda theorem appears in many variations in recent literature. The common denominator is
that the theorem gives a sufficient condition that implies a membership $\phi\in \ideala^l$,
where $\ideala$ is an ideal of some ring $R$.
In the analytic interpretation $R$ is the local ring of an analytic space $Z$, and the condition
is that $|\phi|\leq C|\ideala|^{N+l}$ holds on the space $Z$. The theorem thus relates the rate of vanishing of $\phi$ along the locus of $\ideala$ to actual membership of (powers of) the ideal. The smallest integer $N$ that works for all $\ideala \subset R$ and all $l\geq 1$ simultaneously will be called the Brian\c con-Skoda number of the ring $R$. </p>
<p>The thesis contains three papers. The first one gives an elementary proof of the original Brian\c con-Skoda
theorem. This case is simply $Z=\C^n$.</p>
<p>The second paper contains an analytic proof of a generalization by Huneke. The result is also sharper when $\ideala$ has few generators if the geometry is not to complicated in a certain sense. Moreover, the method can give upper bounds for the Brian\c con-Skoda number for some varieties such as for example the cusp $z^p = w^q$.</p>
<p>In the third paper non-reduced analytic spaces are considered. In this setting Huneke's generalization must be modified to remain valid. More precisely, $\phi$ belongs to $\ideala^l$ if one requires that $|L \phi| \leq C |\ideala|^{N+l}$ holds on $Z$ for a given family of holomorphic differential operators on $Z$.
We impose the assumption that the local ring $\O_Z$ is Cohen-Macaulay for technical reasons.</p>},

publisher={Institutionen för matematiska vetenskaper, Chalmers tekniska högskola,},

place={Göteborg},

year={2010},

series={Preprint - Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University, no: },

note={54},

}

** RefWorks **

RT Dissertation/Thesis

SR Electronic

ID 113072

A1 Sznajdman, Jacob

T1 Some Analytic generalizations of the Briancon-Skoda Theorem

YR 2010

AB <p>The Brian\c con-Skoda theorem appears in many variations in recent literature. The common denominator is
that the theorem gives a sufficient condition that implies a membership $\phi\in \ideala^l$,
where $\ideala$ is an ideal of some ring $R$.
In the analytic interpretation $R$ is the local ring of an analytic space $Z$, and the condition
is that $|\phi|\leq C|\ideala|^{N+l}$ holds on the space $Z$. The theorem thus relates the rate of vanishing of $\phi$ along the locus of $\ideala$ to actual membership of (powers of) the ideal. The smallest integer $N$ that works for all $\ideala \subset R$ and all $l\geq 1$ simultaneously will be called the Brian\c con-Skoda number of the ring $R$. </p>
<p>The thesis contains three papers. The first one gives an elementary proof of the original Brian\c con-Skoda
theorem. This case is simply $Z=\C^n$.</p>
<p>The second paper contains an analytic proof of a generalization by Huneke. The result is also sharper when $\ideala$ has few generators if the geometry is not to complicated in a certain sense. Moreover, the method can give upper bounds for the Brian\c con-Skoda number for some varieties such as for example the cusp $z^p = w^q$.</p>
<p>In the third paper non-reduced analytic spaces are considered. In this setting Huneke's generalization must be modified to remain valid. More precisely, $\phi$ belongs to $\ideala^l$ if one requires that $|L \phi| \leq C |\ideala|^{N+l}$ holds on $Z$ for a given family of holomorphic differential operators on $Z$.
We impose the assumption that the local ring $\O_Z$ is Cohen-Macaulay for technical reasons.</p>

PB Institutionen för matematiska vetenskaper, Chalmers tekniska högskola,

T3 Preprint - Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University, no:

LA eng

LK http://publications.lib.chalmers.se/records/fulltext/113072.pdf

OL 30