## Resolution of singularities in Denjoy-Carleman classes

Venue: | Selecta Math. (N.S |

Citations: | 14 - 1 self |

### BibTeX

@ARTICLE{Bierstone_resolutionof,

author = {Edward Bierstone and Pierre D. Milman},

title = {Resolution of singularities in Denjoy-Carleman classes},

journal = {Selecta Math. (N.S},

year = {},

pages = {1--28}

}

### OpenURL

### Abstract

Abstract. We show that a version of the desingularization theorem of Hironaka holds for certain classes of C ∞ functions (essentially, for subrings that exclude flat functions and are closed under differentiation and the solution of implicit equations). Examples are quasianalytic classes, introduced by E. Borel a century ago and characterized by the Denjoy-Carleman theorem. These classes have been poorly understood in dimension> 1. Resolution of singularities can be used to obtain many new results; for example, topological Noetherianity, ̷Lojasiewicz inequalities, division properties.

### Citations

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(Show Context)
Citation Context ...to obtain many new results; for example, topological Noetherianity, ̷Lojasiewicz inequalities, division properties. 1. Introduction We show that a version of the desingularization theorem of Hironaka =-=[Hi1]-=- holds for certain classes of infinitely differentiable functions – essentially, for subrings that exclude flat functions and are closed under differentiation and the solution of equations satisfying ... |

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Citation Context ...e implicit function theorem. Examples are “quasianalytic classes”, introduced by E. Borel a century ago [Bo1] and characterized (following questions of Hadamard in studies of linear partial equations =-=[Ha]-=-) by the Denjoy-Carleman theorem [De], [Ca]. (See Section 2 below.) Quasianalytic classes in one variable play an important part in harmonic analysis and other areas. (See, for instance, [HJ], [Ko], [... |

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Citation Context ...f functions in a Denjoy-Carleman class in Noetherian. It may seem surprising that desingularization theorems nevertheless hold for Denjoy-Carleman classes. Our proof of resolution of singularities in =-=[BM2]-=-, however, (at least in the case of a hypersurface or “principalization of an ideal” [BM2, Thm. 1.10]) uses only elementary “differential calculus” properties that are satisfied in these classes. This... |

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(Show Context)
Citation Context ... it can be used to prove several other new results about Denjoy-Carleman classes. Many of the geometric properties of semialgebraic sets, for example, are satisfied by o-minimal structures in general =-=[vdDM]-=-. The following are discussed in Section 6 below: (1) Topological Noetherianity (Theorem 6.1). (2) ̷Lojasiewicz inequalities (Theorem 6.2). ProofsBIERSTONE AND MILMAN 3 of ̷Lojasiewicz’s inequalities... |

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Citation Context ...quations [Ha]) by the Denjoy-Carleman theorem [De], [Ca]. (See Section 2 below.) Quasianalytic classes in one variable play an important part in harmonic analysis and other areas. (See, for instance, =-=[HJ]-=-, [Ko], [T].) In several variables, there are beautiful modern developments of E.M. Dyn’kin [Dy1], [Dy2], but the subject is much less understood, perhaps because of a lack of the standard techniques ... |

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(Show Context)
Citation Context ...ersion of resolution of singularities (as in [BM1, Sect. 4]) for quasianalytic classes has already been used by Rolin, Speissegger and Wilkie in their study of o-minimality of Denjoy-Carleman classes =-=[RSW]-=-. In this article, we isolate the properties of a class of C∞ functions needed for resolution of singularities (Section 3). We formulate the most general version of desingularization known for these c... |

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Citation Context ...is divisible by ̂ ξa in the ring of formal power series). Then there exists g ∈ C∞ (W) such that f = ξ · g. Proof. We follow Atiyah’s proof of the division theorem of Hormander and ̷Lojasiewicz. (See =-=[A]-=-, [Hö1], [̷L].) Let ϕ : W ′ → W be a mapping of class C as in Theorem 5.12, such that the pull-back ϕ∗ (ξ) := ξ ◦ϕ is locally a monomial times an invertible factor (in suitable coordinates). Since ϕ∗ ... |

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Citation Context ... a coordinate subspace. If η ∈ C ∞ (U ′ ) is formally a composite with σ, then there exists ζ ∈ C ∞ (U) such that η = σ ∗ (ζ). This assertion is a special case of Glaeser’s composite function theorem =-=[G]-=- since σ is a very simple rational mapping. 7. Proof of the desingularization Theorem 5.12 We begin with a simple but important lemma on transformation of differential operators by blowing up (cf. [H2... |

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Citation Context ... are “quasianalytic classes”, introduced by E. Borel a century ago [Bo1] and characterized (following questions of Hadamard in studies of linear partial equations [Ha]) by the Denjoy-Carleman theorem =-=[De]-=-, [Ca]. (See Section 2 below.) Quasianalytic classes in one variable play an important part in harmonic analysis and other areas. (See, for instance, [HJ], [Ko], [T].) In several variables, there are ... |

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Citation Context ... show that the constants C and D in the proof can be taken more precisely as C = bcm1, D = 1 + bcm1 (cf, [KP, Proposition 1.3.3]).BIERSTONE AND MILMAN 13 Theorem 4.10. (Inverse function theorem; cf. =-=[Kom]-=-.) Let f : U → V denote a Cm-mapping between open subsets U, V of R n . Let x0 ∈ U. Suppose that the Jacobian matrix (∂ϕ/∂x)(x0) is invertible. Then there are neighbourhoods U ′ of x0, V ′ of y0 := f(... |

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Citation Context ...sfies the hypothesis (2.4) sup ( mk+1 mk ) 1/k < ∞ , then C m +1 = Cm, so that Cm is closed under differentiation. (Conversely, if Cm is closed under differentiation, then (2.4) holds (S. Mandelbrojt =-=[M]-=-)). 3. C∞ classes that admit resolution of singularities Suppose that, for every open subset U of Rn , n ∈ N, we have an R-subalgebra C(U) of C∞ (U). Our desingularization thoerems require only the fo... |

7 |
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Citation Context ...15, 14P15, 30D60. The authors’ research was partially supported by NSERC grants OGP0009070, OGP0008949 and the Killam Foundation. 12 RESOLUTION OF SINGULARITIES IN DENJOY-CARLEMAN CLASSES (Childress =-=[Ch]-=-) and it seems unknown (and unlikely) that, in general, a ring of germs of functions in a Denjoy-Carleman class in Noetherian. It may seem surprising that desingularization theorems nevertheless hold ... |

6 |
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Citation Context ...quasianalytic classes”, introduced by E. Borel a century ago [Bo1] and characterized (following questions of Hadamard in studies of linear partial equations [Ha]) by the Denjoy-Carleman theorem [De], =-=[Ca]-=-. (See Section 2 below.) Quasianalytic classes in one variable play an important part in harmonic analysis and other areas. (See, for instance, [HJ], [Ko], [T].) In several variables, there are beauti... |

5 |
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(Show Context)
Citation Context ...lasses in one variable play an important part in harmonic analysis and other areas. (See, for instance, [HJ], [Ko], [T].) In several variables, there are beautiful modern developments of E.M. Dyn’kin =-=[Dy1]-=-, [Dy2], but the subject is much less understood, perhaps because of a lack of the standard techniques of local analytic geometry. For example, the Weierstrass preparation theorem fails 1991 Mathemati... |

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(Show Context)
Citation Context ...]) by the Denjoy-Carleman theorem [De], [Ca]. (See Section 2 below.) Quasianalytic classes in one variable play an important part in harmonic analysis and other areas. (See, for instance, [HJ], [Ko], =-=[T]-=-.) In several variables, there are beautiful modern developments of E.M. Dyn’kin [Dy1], [Dy2], but the subject is much less understood, perhaps because of a lack of the standard techniques of local an... |

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4 |
les séries de polynômes et de fractions rationnelles
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Citation Context ...tives on a “big” set of real line segments in a compact set. If the poles ak accumulate everywhere on such a line segment, we get a quasianalytic function on the line segment that is nowhere analytic =-=[Bo2]-=-. Let C ∞ (U) denote the ring of C ∞ functions on an open subset U of R n . Let f ∈ C ∞ (U). For every α ∈ N n , α = (α1, . . ., αn), we write fα := 1 α! Dα f , where α! = α1! · · ·αn! and D α denotes... |

4 |
The pseudo-analytic extension
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(Show Context)
Citation Context ...in one variable play an important part in harmonic analysis and other areas. (See, for instance, [HJ], [Ko], [T].) In several variables, there are beautiful modern developments of E.M. Dyn’kin [Dy1], =-=[Dy2]-=-, but the subject is much less understood, perhaps because of a lack of the standard techniques of local analytic geometry. For example, the Weierstrass preparation theorem fails 1991 Mathematics Subj... |

1 |
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(Show Context)
Citation Context ...ies the Denjoy-Carleman condition (2.3), then C = Cm and C = ⋃ Cm +j satisfy property (3.4). Our proofs of properties (3.2) and (3.6) are based on a several-variable version of Faà de Bruno’s formula =-=[FdB]-=-. Consider a composite function h = f ◦g, where g(x) = ( g1(x), . . ., gp(x) ) , x = (x1, . . ., xn), and f(y) = f(y1, . . ., yp). Recall that fα(y) denotes Dαf(y)/α!, α ∈ Np . Write gγ := (g1,γ, . . ... |

1 |
Ultradistributions définies sur IRn et sur certaines classes de variétés différentiables
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(Show Context)
Citation Context ...t all |δi| are distinct because if |δi| = |δj| for some i and j ̸= i, then we can replace m |ki| |δi| m|kj| |δj| in the left-hand side of (4.6) by m |ki|+|kj| |δi| .) □ Theorem 4.7. (Composition; cf. =-=[Rou]-=-.) Let U and V denote open subsets of R n and R p , respectively. Let f ∈ Cm(V ) and let g = (g1, . . ., gp) : U → V , where each gj ∈ Cm(U). Then f ◦g ∈ Cm(U). Proof. Let K be a compact subset of U. ... |

1 |
The ̷Lojasiewicz inequality for very smooth functions, Soviet
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- 1990
(Show Context)
Citation Context ... resolution of singularities were already given in [BM2, Sect.2]. These inequalities were known previously for Denjoy-Carleman classes only in dimension 2, under more restrictive hypotheses (Vol’berg =-=[V]-=-). (3) Division properties (Theorem 6.3). Several unresolved questions about Denjoy-Carleman classes are raised in the text. We are grateful to Vincent Thilliez for clarifying many points about quasia... |