Torification and factorization of birational maps

Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; Włodarczyk, Jarosław

doi:10.1090/s0894-0347-02-00396-x

Cited by 262 publications

(376 citation statements)

References 73 publications

(100 reference statements)

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“…Since ϕ i is trivial over R >0 (resp. R <0 ) and χ c (a ≥ 0, b = 0) = 0, we finally have χ c F > ([ϕ 1 : X 1 → R * ] * [ϕ 2 : X 2 → R * ]) = −χ c ϕ 1 > 0,ϕ 2 > 0 = −χ c F > [ϕ 1 : 2…”

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confidence: 91%

On a motivic invariant of the arc-analytic equivalence

Campesato¹

2017

Annales De l'Institut Fourier

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To a Nash function germ, we associate a zeta function similar to the one introduced by J. Denef and F. Loeser. Our zeta function is a formal power series with coefficients in the Grothendieck ring $\mathcal{M}$ of $\mathcal{AS}$-sets up to $\mathbb{R}^*$-equivariant $\mathcal{AS}$-bijections over $\mathbb{R}^*$, an analog of the Grothendieck ring constructed by G. Guibert, F. Loeser and M. Merle. This zeta function generalizes the previous construction of G. Fichou but thanks to its richer structure it allows us to get a convolution formula and a Thom-Sebastiani type formula. We show that our zeta function is an invariant of the arc-analytic equivalence, a version of the blow-Nash equivalence of G. Fichou. The convolution formula allows us to obtain a partial classification of Brieskorn polynomials up to the arc-analytic equivalence by showing that the exponents are arc-analytic invariants

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confidence: 91%

On a motivic invariant of the arc-analytic equivalence

Campesato¹

2017

Annales De l'Institut Fourier

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“…We label the transformation from (2.9) by w 3 and the corresponding operation using points p 5 and p 7 by w 2 . These transformations and two natural symmetries, ρ 1 and ρ 2 , form a representation of an affine Weyl group of type D (1) 5 (see [47,Section 2] for more details). Furthermore, as an infinite order isomorphism, both (2.5) and (2.6) may be represented as a product of these involutions as…”

Section: U-p Amentioning

confidence: 99%

From Polygons to Ultradiscrete Painlevé Equations

Ormerod

Yamada

2015

SIGMA

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Abstract. The rays of tropical genus one curves are constrained in a way that defines a bounded polygon. When we relax this constraint, the resulting curves do not close, giving rise to a system of spiraling polygons. The piecewise linear transformations that preserve the forms of those rays form tropical rational presentations of groups of affine Weyl type. We present a selection of spiraling polygons with three to eleven sides whose groups of piecewise linear transformations coincide with the Bäcklund transformations and the evolution equations for the ultradiscrete Painlevé equations.

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“…We recall the main properties of the collection AS in subsection 2. 1. The virtual Poincaré polynomial was constructed by C. McCrory, A.…”

Section: Introductionmentioning

confidence: 99%

An Inverse Mapping Theorem for Blow-Nash Maps on Singular Spaces

Campesato¹

2016

Nagoya Math. J.

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A semialgebraic map f : X → 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 → 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 Poincaré polynomial of McCrory-Parusiński 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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Torification and factorization of birational maps

Cited by 262 publications

References 73 publications

On a motivic invariant of the arc-analytic equivalence

On a motivic invariant of the arc-analytic equivalence

From Polygons to Ultradiscrete Painlevé Equations

An Inverse Mapping Theorem for Blow-Nash Maps on Singular Spaces

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