Stable polynomial division and essential normality of graded Hilbert modules

Abstract

The purpose of this paper is to initiate a new attack on Arveson's resistant conjecture, that all graded submodules of the $d$-shift Hilbert module $H^2$ are essentially normal. We introduce the stable division property for modules (and ideals): a normed module $M$ over the ring of polynomials in $d$ variables has the stable division property if it has a generating set $\{f_1,...,f_k\}$ such that every $h\in M$ can be written as $h={\sum }_i{a}_i{f}_i$ for some polynomials $a_i$ such that $\sum \Vert a_i{f}_i\Vert \le C\Vert h\Vert$. We show that certain classes of modules have this property, and that the stable decomposition $h=\sum a_i{f}_i$ may be obtained by carefully applying techniques from computational algebra. We show that when the algebra of polynomials in $d$ variables is given the natural ${\ell }^1$ norm, then every ideal is linearly equivalent to an ideal that has the stable division property. We then show that a module $M$ that has the stable division property (with respect to the appropriate norm) is $p$-essentially normal for $p>\dim (M)$, as conjectured by Douglas. This result is used to give a new, unified proof that certain classes of graded submodules are essentially normal. Finally, we reduce the problem of determining whether all graded submodules of the $d$-shift Hilbert module are essentially normal, to the problem of determining whether all ideals generated by quadratic scalar valued polynomials are essentially normal.

17 pages. Minor changes, close to published version

References in corpus (6)

• What can be computed in algebraic geometry?
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• Essentially Reductive Hilbert Modules II
• Stable normal forms for polynomial system solving
• Essentially Reductive Hilbert Modules
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