Algebraic approximation preserving dimension

Ferrarotti, Massimo; Fortuna, Elisabetta; Wilson, Leslie C.

doi:10.1007/s10231-016-0583-6

Cited by 3 publications

(3 citation statements)

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“…In the papers [FFW1], [FFW2], [FFW3] and [FFW4] we addressed the question of the existence of an algebraic representative Y in the class of s-equivalence of a given subanalytic set A at a fixed point P . In this case we also say that Y s-approximates A.…”

Section: Introductionmentioning

confidence: 99%

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Tangential approximation of analytic sets

Ferrarotti,

Fortuna,

Wilson

2019

Preprint

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Two subanalytic subsets of R n are called s-equivalent at a common point P if the Hausdorff distance between their intersections with the sphere centered at P of radius r vanishes to order > s as r tends to 0. In this work we strengthen this notion in the case of real subanalytic subsets of R n with isolated singular points, introducing the notion of tangential s-equivalence at a common singular point which considers also the distance between the tangent planes to the sets near the point. We prove that, if V (f ) is the zero-set of an analytic map f and if we assume that V (f ) has an isolated singularity, say at the origin O, then for any s ≥ 1 the truncation of the Taylor series of f of sufficiently high order defines an algebraic set with isolated singularity at O which is tangentially s-equivalent to V (f ).

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Section: Introductionmentioning

confidence: 99%

“…The answer to the previous question is in general negative for subanalytic sets. However, for any real number s ≥ 1 and for any closed semianalytic set A ⊂ R n of codimension ≥ 1, there exists an algebraic subset Y of R n such that A ∼ s Y (and Y can be chosen of the same dimension as A; see [FFW3] and [FFW4]).…”

Section: Introductionmentioning

confidence: 99%

Tangential approximation of analytic sets

Ferrarotti,

Fortuna,

Wilson

2019

Preprint

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“…e.g. [5], [6], [10], [12], [11]. But the classical methods of such approximation do not provide the objects which are homeomorphic.…”

Section: Introductionmentioning

confidence: 99%

Higher order approximation of analytic sets by topologically equivalent algebraic sets

et al. 2017

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It is known that every germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that the homeomorphism can be chosen in such a way that the analytic and algebraic germs are tangent with any prescribed order of tangency. Moreover, the space of arcs contained in the algebraic germ approximates the space of arcs contained in the analytic one, in the sense that they are identical up to a prescribed truncation order.2010 Mathematics Subject Classification. 32S05, 32S15, 13B40.

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