Families of polynomials and their specializations

Bodin, Arnaud; Dèbes, Pierre; Najib, Salah

doi:10.1016/j.jnt.2016.06.023

Cited by 6 publications

(4 citation statements)

References 21 publications

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“…A circle γ that is not fully contained in Z(f ) crosses it in at most O(D) points, which follows from Bézout's theorem (see, e.g., [3]). This yields a total of O(nD) = O(m 3/5 n 3/5 + m) incidences, within the asymptotic bound as in (2). It therefore remains to bound the number of incidences between the points of P on Z(f ) and the anchored circles that are fully contained in Z(f ).…”

Section: Proof Of Theorem 21mentioning

confidence: 85%

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Incidences between points and curves with almost two degrees of freedom

Sharir¹,

Solomon²,

Zlydenko³

2020

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We study incidences between points and (constant-degree algebraic) curves in three dimensions, taken from a family C of curves that have almost two degrees of freedom, meaning that (i) every pair of curves of C intersect in O(1) points, (ii) for any pair of points p, q, there are only O(1) curves of C that pass through both points, and (iii) a pair p, q of points admit a curve of C that passes through both of them if and only if F (p, q) = 0 for some polynomial F of constant degree associated with the problem. (As an example, the family of unit circles in R 3 that pass through some fixed point is such a family.)We begin by studying two specific instances of this scenario. The first instance deals with the case of unit circles in R 3 that pass through some fixed point (so called anchored unit circles). In the second case we consider tangencies between directed points and circles in the plane, where a directed point is a pair (p, u), where p is a point in the plane and u is a direction, and (p, u) is tangent to a circle γ if p ∈ γ and u is the direction of the tangent to γ at p. A lifting transformation due to Ellenberg et al. [4] maps these tangencies to incidences between points and curves ('lifted circles') in three dimensions. In both instances we have a family of curves in R 3 with almost two degrees of freedom.We show that the number of incidences between m points and n anchored unit circles in R 3 , as well as the number of tangencies between m directed points and n arbitrary circles in the plane, is O(m 3/5 n 3/5 + m + n) in both cases.We then derive a similar incidence bound, with a few additional terms, for more general families of curves in R 3 with almost two degrees of freedom, under a few additional natural assumptions.The proofs follow standard techniques, based on polynomial partitioning, but they face a critical novel issue involving the analysis of surfaces that are infinitely ruled by the respective family of curves, as well as of surfaces in a dual three-dimensional space that are infinitely ruled by the respective family of suitably defined dual curves. We either show that no such surfaces exist, or develop and adapt techniques for handling incidences on such surfaces.The general bound that we obtain is O(m 3/5 n 3/5 + m + n) plus additional terms that depend on how many curves or dual curves can lie on an infinitely-ruled surface.

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Section: Proof Of Theorem 21mentioning

confidence: 85%

“…We are indebted to Noam Solomon, who has observed that, using some (advanced) algebraic geometry theory, based on the Noether-Bertini theorem (see[2,14]), one can show that for most points p, except for those that lie in some lower-dimensional variety, p * is indeed a curve.…”

mentioning

confidence: 99%

Incidences between points and curves with almost two degrees of freedom

Sharir¹,

Solomon²,

Zlydenko³

2020

Preprint

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“…Lemma 3.1 is our specialization tool here. Based on results of Bertini, Krull and Noether, it is in the same vein as those from [BDN09], [BDN17]. We prove it in §3.1, then deduce Theorem 1.4 in §3.2.…”

Section: The Geometric Partmentioning

confidence: 55%

The Schinzel Hypothesis for Polynomials

Bodin¹,

Dèbes²,

Najib³

2019

Preprint

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The Schinzel hypothesis is a famous conjectural statement about primes in value sets of polynomials, which generalizes the Dirichlet theorem about primes in an arithmetic progression. We consider the situation that the ring of integers is replaced by a polynomial ring and prove the Schinzel hypothesis for a wide class of them: polynomials in at least one variable over the integers, polynomials in several variables over an arbitrary field, etc. We achieve this goal by developing a version over rings of the Hilbert specialization property. A polynomial Goldbach conjecture is deduced, along with a result on spectra of rational functions.

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“…If 𝑓 ∈ 𝐾 (𝑋, 𝑌 ) is non-composite then 𝜌( 𝑓 ) < (deg 𝑓 ) 2 + deg 𝑓 . Some variants of these results have been extensively studied; see for example [2], [3], [4, Section 5.5], [5], [14] and [15].…”

Section: 𝜎( 𝑓mentioning

confidence: 99%