[Paper Review] A counterexample concerning quantifier elimination in quasianalytic structures

Krzysztof Jan Nowak|arXiv (Cornell University)|Oct 4, 2013

Advanced Topology and Set Theory16 references3 citations

TL;DR

This paper constructs a counterexample showing that quasianalytic structures, unlike the classical real analytic structure ℝ_an, do not admit quantifier elimination when augmented only by the reciprocal function 1/x. Using rectilinearization of terms and non-extendability theorems for Denjoy–Carleman classes, it proves that certain definable curves cannot be expressed in this language, and further shows Łojasiewicz’s theorem on subanalytic curves being semianalytic fails in this setting.

ABSTRACT

This paper is a revised version of our preprints IMUJ Preprint 2012/04 and RAAG Preprint 343 from May 2012. It provides an example of a quasianalytic structure which, unlike the classical analytic structure, does not admit quantifier elimination in the language of restricted quasianalytic functions augmented by the reciprocal function. It also demonstrates that Lojasiewicz's theorem that every subanalytic curve is semianalytic is no longer true in quasianalytic structures. Our construction applies rectilinearization of terms, established in our earlier papers, as well as some theorems on power substitution for Denjoy-Carleman classes and on non-extendability of quasianalytic function germs. The last result relies on Grothendieck's factorization and open mapping theorems for (LF)-spaces.

Motivation & Objective

• To resolve an open problem from earlier work regarding whether quasianalytic structures admit quantifier elimination in the language of restricted quasianalytic functions augmented by 1/x.
• To demonstrate that unlike ℝ_an, such quasianalytic structures may not eliminate quantifiers in this language.
• To show that Łojasiewicz’s theorem—every subanalytic curve is semianalytic—fails in quasianalytic settings.
• To establish that certain definable curves in quasianalytic structures cannot be parametrized using only rational functions and restricted quasianalytic functions.
• To provide a non-extendability result for quasianalytic function germs using Grothendieck’s theorems on (LF)-spaces.

Proposed method

• Utilizes rectilinearization of terms via blow-ups and power substitutions in quasianalytic structures, building on prior results in [18] and [19].
• Applies power substitution theorems for Denjoy–Carleman classes to analyze the local behavior of functions near singularities.
• Employs a non-extendability theorem for quasianalytic germs, refined from Thilliez’s work, relying on Grothendieck’s factorization and open mapping theorems for (LF)-spaces.
• Constructs a specific quasianalytic class 𝒬_M using a log-convex sequence M_n = (log log 3)^{-3} (log log(n+3))^{n+3}, ensuring quasianalyticity and closure under derivatives.
• Uses Puiseux-type parametrization to derive a contradiction: if the curve were definable in the restricted language, its parametrization would imply membership in a smaller quasianalytic class.
• Applies the rectilinearization theorem (Theorem 4.2) to show that any definable function in the language must arise from a composition of blow-ups and power substitutions, leading to a contradiction if the function lies outside the expected class.

Experimental results

Research questions

• RQ1 Does a quasianalytic structure admit quantifier elimination in the language of restricted quasianalytic functions augmented only by the reciprocal function 1/x?
• RQ2 Can every definable curve in a quasianalytic structure be parametrized using only rational functions and restricted quasianalytic functions?
• RQ3 Is Łojasiewicz’s theorem—that every subanalytic set of dimension ≤1 is semianalytic—valid in quasianalytic structures?
• RQ4 Are there quasianalytic function germs that cannot be extended to larger domains, even locally, in a way compatible with the structure’s definable functions?
• RQ5 Can rectilinearization via blow-ups and power substitutions fully describe the definable sets in such structures when only 1/x is added?

Key findings

• There exists a quasianalytic structure, specifically ℝ_𝒬_M with M_n = (log log 3)^{-3} (log log(n+3))^{n+3}, that does not admit quantifier elimination in the language augmented by 1/x.
• The counterexample is a plane curve through the origin, definable in ℝ_𝒬_M, which cannot be defined by any term in the language of restricted 𝒬_M-functions and 1/x.
• This failure arises because the curve’s germ at 0 lies in 𝒬_1(M)^+ but not in any ⋃_{p odd} 𝒬_1(M^{(p)}), contradicting the necessary parametrization form for definability.
• The result implies that Łojasiewicz’s theorem on subanalytic curves being semianalytic does not hold in quasianalytic structures.
• The non-extendability of certain quasianalytic germs is established using Grothendieck’s theorems on (LF)-spaces, showing that such germs cannot be represented in the restricted language.
• The construction confirms that rectilinearization via blow-ups alone (without rational powers) is insufficient to describe all definable sets in quasianalytic structures, even in dimension 1.