An inverse mapping theorem for blow-Nash maps on singular spaces

Jean-Baptiste Campesato

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An inverse mapping theorem for blow-Nash maps on singular spaces

Jean-Baptiste Campesato

Algebraic Geometry · Submitted 25 Jun 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that if a blow-Nash self-homeomorphism $f:X\rightarrow X$ satisfies a lower bound of the Jacobian determinant condition then $f^{-1}$ is also blow-Nash and satisfies the same condition.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

A semialgebraic map $f:X\to Y$ between two real algebraic sets is called blow-Nash if it can be made Nash (i.e. semialgebraic and real analytic) by composing with finitely many blowings-up with non-singular centers. We prove that if a blow-Nash self-homeomorphism $f:X\rightarrow X$ satisfies a lower bound of the Jacobian determinant condition then $f^{-1}$ is also blow-Nash and satisfies the same condition. The proof relies on motivic integration arguments and on the virtual Poincar\'e polynomial of McCrory-Parusi\'nski and Fichou. In particular, we need to generalize Denef-Loeser change of variables key lemma to maps that are generically one-to-one and not merely birational.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A semialgebraic map $f:X\to Y$ between two real algebraic sets is called blow-Nash if it can be made Nash (i.e. semialgebraic and real analytic) by composing with finitely many blowings-up with non-singular centers.	background 1.00	0.06	0.02	asserted 1.00
s1	We prove that if a blow-Nash self-homeomorphism $f:X\rightarrow X$ satisfies a lower bound of the Jacobian determinant condition then $f^{-1}$ is also blow-Nash and satisfies the same condition.	result 0.96	0.96	0.94	asserted 1.00
s2	The proof relies on motivic integration arguments and on the virtual Poincar\'e polynomial of McCrory-Parusi\'nski and Fichou.	method 1.00	0.43	0.02	asserted 1.00
s3	In particular, we need to generalize Denef-Loeser change of variables key lemma to maps that are generically one-to-one and not merely birational.	method 0.50	0.84	0.03	asserted 0.99

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Original abstract

A semialgebraic map $f:X\to Y$ between two real algebraic sets is called blow-Nash if it can be made Nash (i.e. semialgebraic and real analytic) by composing with finitely many blowings-up with non-singular centers. We prove that if a blow-Nash self-homeomorphism $f:X\rightarrow X$ satisfies a lower bound of the Jacobian determinant condition then $f^{-1}$ is also blow-Nash and satisfies the same condition. The proof relies on motivic integration arguments and on the virtual Poincar\'e polynomial of McCrory-Parusi\'nski and Fichou. In particular, we need to generalize Denef-Loeser change of variables key lemma to maps that are generically one-to-one and not merely birational.

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1 arXiv paper cite this in 4 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Moreover, by [ref] , $φ̃$ induces a bijection between arcs on $M$ and arcs on $ℝ^d$ not entirely included in some nowhere dense subset of $ℝ^d$.

On a motivic invariant of the arc-analytic equivalence · Arc-analytic equivalence · 0.98

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We refer the reader to [ref] for the properties of $ℒ_n(M)$ and $ℒ(M)$.

On a motivic invariant of the arc-analytic equivalence · Definition · 0.95

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The argument of the following lemma is essentially the same as [ref] .

Inverse Function Theorems for Arc-analytic Homeomorphisms · Essence of the proof · 0.87

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