Motivic invariants of real polynomial functions and their Newton polyhedrons

Fichou, Goulwen; Fukui, Toshizumi

doi:10.1017/s030500411500064x

Cited by 2 publications

(4 citation statements)

References 19 publications

(51 reference statements)

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“…A similar formula to the one of the following result has been proved in different settings: by Varchenko for the zeta function of the monodromy, by Denef and Loeser [, § 5] for the topological zeta function, by Denef and Hoornaert [, Theorem 4.2] for the Igusa

p

‐adic zeta function and by Guibert for the motivic zeta function [, Proposition 2.1.3]. The proof of Fichou and Fukui already relies on an adaptation of this construction to the virtual Poincaré polynomial. Theorem Let

f \in R [x_{1}, \dots, x_{d}]

be non‐degenerate, then the following equality holds in

\hat{scriptM} ⟦ T ⟧

\begin{matrix} true \\ right Z_{f} (T) & = & left \sum_{τ \in {normalΓ}_{f}^{c}} (([] f_{τ} : {({double-struckR}^{*})}^{d} ∖ f_{τ}^{- 1} false(0 false) \to {double-struckR}^{*}) + ([] {prefixpr}_{2} : f_{τ}^{- 1} false(0 false) \cap {({double-struckR}^{*})}^{d} \times {double-struckR}^{*} \to {double-struckR}^{*}) \frac{L^{- 1} T}{double-struck1 - L^{- 1} T}) \\ left \times S_{σ false(τ false)} (T), \end{matrix}

where

R^{*}

acts on

f_{τ}^{- 1} false(0 false) \cap {({double-struckR}^{*})}^{d} \times {double-struckR}^{*}

λ \cdot (x, t

…”

Section: The Modified Zeta Function Of a Non‐degenerate Polynomialmentioning

confidence: 72%

“…mon where the action is given by λ A similar formula to the one of the following result has been proved in different settings: by A. N. Varchenko [29] for the zeta function of the monodromy, by J. Denef and F. Loeser [6, §5] for the topological zeta function, by J. Denef and K. Hoornaert [5, Theorem 4.2] for the Igusa p-adic zeta function and by G. Guibert for the motivic zeta function [14, Proposition 2.1.3]. The proof of G. Fichou and T. Fukui [12] already relies on an adaptation of this construction to the virtual Poincaré polynomial.…”

Section: Proposition 212 ([3 End Of §3]) the Map Asmentioning

confidence: 99%

“…Then G. Fichou and T. Fukui [12] proved the conjecture for non-degenerate convenient weighted homogeneous polynomials in three variables up to the blow-Nash equivalence.…”

Section: Introductionmentioning

confidence: 98%

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On the arc-analytic type of some weighted homogeneous polynomials

Campesato

2018

J. London Math. Soc.

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It is known that the weights of a complex weighted homogeneous polynomial f with isolated singularity are analytic invariants of (C d , f −1 (0)). When d = 2, 3 this result holds by assuming merely the topological type instead of the analytic one.Fichou and Fukui recently proved the following real counterpart: the blow-Nash type of a real singular non-degenerate convenient weighted homogeneous polynomial in three variables determines its weights.The aim of this paper is to generalize the above-cited result with no condition on the number of variables. We work with a characterization of the blow-Nash equivalence called the arc-analytic equivalence. It is an equivalence relation on Nash function germs with no continuous moduli which may be seen as a semialgebraic version of the blow-analytic equivalence of Kuo. Contents

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p

f \in R [x_{1}, \dots, x_{d}]

be non‐degenerate, then the following equality holds in

\hat{scriptM} ⟦ T ⟧

\begin{matrix} true \\ right Z_{f} (T) & = & left \sum_{τ \in {normalΓ}_{f}^{c}} (([] f_{τ} : {({double-struckR}^{*})}^{d} ∖ f_{τ}^{- 1} false(0 false) \to {double-struckR}^{*}) + ([] {prefixpr}_{2} : f_{τ}^{- 1} false(0 false) \cap {({double-struckR}^{*})}^{d} \times {double-struckR}^{*} \to {double-struckR}^{*}) \frac{L^{- 1} T}{double-struck1 - L^{- 1} T}) \\ left \times S_{σ false(τ false)} (T), \end{matrix}

where

R^{*}

acts on

f_{τ}^{- 1} false(0 false) \cap {({double-struckR}^{*})}^{d} \times {double-struckR}^{*}

λ \cdot (x, t

…”

Section: The Modified Zeta Function Of a Non‐degenerate Polynomialmentioning

confidence: 72%

Section: Proposition 212 ([3 End Of §3]) the Map Asmentioning

confidence: 99%

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On the arc-analytic type of some weighted homogeneous polynomials

Campesato

2018

J. London Math. Soc.

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show abstract

“…Case of a weighted homogeneous polynomial. In this section, we state a formula for the local motivic Milnor fibres associated with a polynomial function non-degenerate with respect to its Newton polyhedron (we will use such results in section 4), using results in [17]. We begin with some notation.…”

Section: Motivic Milnor Fibres With Signmentioning

confidence: 99%