A property of ideals of jets of functions vanishing on a set

Charles Fefferman; Ary Shaviv

doi:10.4171/RMI/1423

A property of ideals of jets of functions vanishing on a set

Charles Fefferman, Ary Shaviv

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

For a set E ⊂ Rⁿ that contains the origin, we consider I ^m(E) – the set of all m^th degree Taylor approximations (at the origin) of C ^m functions on Rⁿ that vanish on E. This set is a proper ideal in P ^m(Rⁿ) – the ring of all m^th degree Taylor approximations of C ^m functions on Rⁿ. Which ideals in P ^m(Rⁿ) arise as I ^m(E) for some E? In this paper we introduce the notion of a closed ideal in P ^m(Rⁿ), and prove that any ideal of the form I ^m(E) is closed. We do not know whether in general any closed proper ideal is of the form I ^m(E) for some E, however we prove in a subsequent paper that all closed proper ideals in P ^m(Rⁿ) arise as I ^m(E) when m C n ≤ 5.

Original language	English (US)
Pages (from-to)	719-752
Number of pages	34
Journal	Revista Matematica Iberoamericana
Volume	40
Issue number	2
DOIs	https://doi.org/10.4171/RMI/1423
State	Published - 2024

All Science Journal Classification (ASJC) codes

General Mathematics

Keywords

Whitney extension problems
algorithms
closed ideals
differentiable functions
extrapolation
ideals of jets
jets of functions
real functions
semi-algebraic sets

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10.4171/RMI/1423

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abstract = "For a set E ⊂ Rn that contains the origin, we consider I m(E) – the set of all mth degree Taylor approximations (at the origin) of C m functions on Rn that vanish on E. This set is a proper ideal in P m(Rn) – the ring of all mth degree Taylor approximations of C m functions on Rn. Which ideals in P m(Rn) arise as I m(E) for some E? In this paper we introduce the notion of a closed ideal in P m(Rn), and prove that any ideal of the form I m(E) is closed. We do not know whether in general any closed proper ideal is of the form I m(E) for some E, however we prove in a subsequent paper that all closed proper ideals in P m(Rn) arise as I m(E) when m C n ≤ 5.",

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AU - Shaviv, Ary

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KW - Whitney extension problems

KW - algorithms

KW - closed ideals

KW - differentiable functions

KW - extrapolation

KW - ideals of jets

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KW - real functions

KW - semi-algebraic sets

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A property of ideals of jets of functions vanishing on a set

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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