On the Briancon-Skoda theorem on a singular variety

关于叶状结构的Briançon-Skoda定理

On Briançon-Skoda theorem for foliations

本文将Briançon-Skoda定理推广到第二型叶状结构，建立其Milnor数与Tjurina数的关系，并给出代数曲线全局Tjurina数的下界。

Briançon-Skoda exponents and the maximal root of reduced Bernstein-Sato polynomials

For a holomorphic function $f$ on a complex manifold $X$, the Briançon-Skoda exponent $e^{\rm BS}(f)$ is the smallest integer $k$ with $f^k\in(\partial f)$ (replacing $X$ with a neighborhood of $f^{-1}(0)$), where $(\partial f)$ denotes the Jacobian ideal of $f$. It is shown that $e^{\rm BS}(f)\le d_X$ $(:=\dim X)$ by Brian\c con-Skoda. We prove that $e^{\rm BS}(f)\le[d_X-2\widetildeα_f]+1$ with $-\widetildeα_f$ the maximal root of the reduced Bernstein-Sato polynomial $b_f(s)/(s+1)$, assuming the latter exists (shrinking $X$ if necessary). This implies for instance that $e^{\rm BS}(f)\le d_X-…

A Briançon-Skoda type result for a non-reduced analytic space

We present here an analogue of the Briançon-Skoda theorem for a germ of an analytic space $Z$ at 0, such that $O_{Z,0}$ is Cohen-Macaulay, but not necessarily reduced. More precisely, we find a sufficient condition for membership of a function in a power of an arbitrary ideal $a^l \subset O_{Z,0}$ in terms of size conditions of Noetherian differential operators applied to that function. This result generalizes a theorem by Huneke in the reduced case.

Explicit versions of the Briançon-Skoda theorem with variations

We give new a proof of the general Briançon-Skoda theorem about ideals of holomorphic functions by means of multivariable residue calculus. The method gives new variants of this theorem for products of ideals. Moreover, we obtain a related result for the ideal generated by the the subdeterminants of a matrix-valued generically surjective holomorphic function, generalizing the duality theorem for a complete intersection. We also provide explicit versions of the various results, including the general Briançon-Skoda theorem, with integral representation formulas.

The Briancon-Skoda number of analytic irreducible planar curves

The Briancon-Skoda number of a ring R is defined as the smallest integer k, such that for any ideal I\subset R and r\geq 1, the integral closure of I^{k+r-1} is contained in I^r. We compute the Briancon-Skoda number of the local ring of any analytic irreducible planar curve in terms of its Puiseux characteristics. It turns out that this number is closely related to the Milnor number.

An elementary proof of the Briancon-Skoda theorem

We give a new elementary proof of the Briançon-Skoda theorem, which states that for an $m$-generated ideal $\mathfrak{a}$ in the ring of germs of analytic functions at $0\in \C^n$, the $ν$:th power of its integral closure is contained in $\mathfrak{a}$, where $ν= \min(m,n)$.