On the number of solutions of an algebraic equation on the curve 𝑦=𝑒^{𝑥}+sin𝑥,𝑥&gt;0, and a consequence for o-minimal structures

Gwoździewicz, Janusz; Kurdyka, Krzysztof; Parusiński, Adam

doi:10.1090/s0002-9939-99-04672-9

Cited by 9 publications

(10 citation statements)

References 6 publications

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“…The construction of the curves Z dm . The construction from this section is inspired by [GKP99,BLN19].…”

Section: Proof Of Proposition 21mentioning

confidence: 99%

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Instability of Bézout's Theorem and arbitrarily large intersection between symplectic surfaces

Ancona¹,

Lerário²

2021

Preprint

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We investigate the failure of Bézout's Theorem for two symplectic surfaces in CP 2 (and more generally on an algebraic surface), by proving that every plane algebraic curve C can be perturbed in the C 1 -topology to an arbitrarily close smooth symplectic surface C with the property that the cardinality #C ∩ Z d of the transversal intersection of C with an algebraic plane curve Z d of degree d, as a function of d can grow arbitrarily fast. As a consequence we obtain that, although Bézout's Theorem is true for pseudoholomorphic curves with respect to the same almost complex structure, it is "arbitrarily false" for pseudoholomorphic curves with respect to different (but arbitrarily close) almost-complex structures (we call this phenomenon "instability of Bézout's Theorem").

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“…The construction of the curves Z dm . The construction from this section is inspired by [GKP99,BLN19].…”

Section: Proof Of Proposition 21mentioning

confidence: 99%

“…In particular, no quantitative estimate can be expected even in the tame world. For a "real" counterpart of this problem, see [BLN19,GKP99].…”

Section: Introductionmentioning

confidence: 99%

Instability of Bézout's Theorem and arbitrarily large intersection between symplectic surfaces

Ancona¹,

Lerário²

2021

Preprint

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“…A result can be formulated for any real analytic (or even smooth) function f with suitable finiteness properties (zeros of derivatives, derivatives of the inverse, and algebraic relations). An example of such a function that is not Pfaffian is exhibited in [4]. (Indeed, the given example e x + sin x does not belong to any o-minimal structure (see [4]). )…”

Section: Remark 33mentioning

confidence: 99%

“…An example of such a function that is not Pfaffian is exhibited in [4]. (Indeed, the given example e x + sin x does not belong to any o-minimal structure (see [4]). ) Remark 3.4.…”

Section: Remark 33mentioning

confidence: 99%

Note on the Rational Points of a Pfaff Curve

Pila

2006

Proceedings of the Edinburgh Mathematical Society

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“…While the polynomial method has yielded impressive results, its reliance on Bézout's Theorem limits its scope to questions about algebraic and semialgebraic sets. If one tries to generalize it to sets definable in o-minimal structures other than real closed fields, Bézout's theorem can fail [16]. The cutting lemma method, however, can be generalized to more complicated sets using the language of model theory.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Bounds in Distal Structures

Anderson

2021

Preprint

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We provide polynomial upper bounds for the minimal sizes of distal cell decompositions in several kinds of distal structures, particularly weakly o-minimal and P -minimal structures. The bound in general weakly o-minimal structures generalizes the vertical cell decomposition for semialgebraic sets, and the bounds for vector spaces in both o-minimal and p-adic cases are tight. We apply these bounds to Zarankiewicz's problem and sum-product bounds in distal structures. PreliminariesIn this section, we review the notation and model-theoretic framework necessary to understand distal cell decompositions. For further background on these definitions, see [4] and [5].Firstly, we review asymptotic notation:• We will say f (x) = Ω(g(x)) to indicate that there existsThroughout this section, let M be a first-order structure in the language L. We will frequently refer to Φ(x; y) as a set of formulas, which will implicitly be in the language L. Each formula in Φ will have the same variables, split into a tuple x and a tuple y, where, for instance, |x| represents the length of the tuple x. We use M to refer to the universe, or underlying set, of M, and M n to refer to its nth Cartesian power. If φ(x; y) is a formula with its variables partitioned into x and y, and b ∈ M |y| , then φ(M |x| ; b) refers to the definable set {a ∈ M |x| : M |= φ(a, b)}. We also define the dual formula of φ(x; y) to be φ * (y; x) such that M |= ∀x∀yφ(x; y) ↔ φ * (y; x), and similarly define Φ * (y; x) to be the set {φ * (y; x) : φ(x; y) ∈ Φ(x; y)}.

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On the number of solutions of an algebraic equation on the curve 𝑦=𝑒^{𝑥}+sin𝑥,𝑥>0, and a consequence for o-minimal structures

Cited by 9 publications

References 6 publications

Instability of Bézout's Theorem and arbitrarily large intersection between symplectic surfaces

Instability of Bézout's Theorem and arbitrarily large intersection between symplectic surfaces

Note on the Rational Points of a Pfaff Curve

Combinatorial Bounds in Distal Structures

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