Squarefree polynomials with prescribed coefficients

Oppenheim, Amotz; Shusterman, Mark

doi:10.1016/j.jnt.2017.10.025

Cited by 4 publications

(3 citation statements)

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“…Throughout mathematics, the equidistribution properties of certain objects are a central theme studied by many authors, including in areas of algebraic and arithmetic geometry [17,26,32] and number theory [44,50]. Recently, there has been a body of work in analogy with Dirichlet's theorem on the asymptotic equidistribution (or non-equidistribution) on arithmetic progressions of various objects.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Asymptotic equidistribution for partition statistics and topological invariants

Giulia¹,

Craig²,

Males³

2021

Preprint

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We provide a general framework for proving asymptotic equidistribution, convexity, and logconcavity of coefficients of generating functions on arithmetic progressions. Our central tool is a variant of Wright's Circle Method proved by two of the authors with Bringmann and Ono, following work of Ngo and Rhoades. We offer a selection of different examples of such results, proving asymptotic equidistribution results for several partition statistics, modular sums of Betti numbers of two-and three-flag Hilbert schemes, and the number of cells of dimension a (mod b) of a certain scheme central in work of Göttsche.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Asymptotic equidistribution for partition statistics and topological invariants

Giulia¹,

Craig²,

Males³

2021

Preprint

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“…In addition, we remark that, by a recent result of Oppenheim and Shusterman [14,Theorem 1.2], for any polynomial f ∈ F p [x] of degree d ≥ 2, there exists a square-free polynomial g of degree d such that L p (f − g) ≤ 2(d − 1).…”

Section: Polynomials Over Prime Fieldsmentioning

confidence: 91%

The distance to square-free polynomials

Dubickas¹,

Sha²

2018

Acta Arith.

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In this paper, we consider a variant of Turán's problem on the distance from an integer polynomial in Z[x] to the nearest irreducible polynomial in Z [x]. We prove that for any polynomial f ∈ Z[x], there exist infinitely many square-free polynomials g ∈ Z [x] such that L(f − g) ≤ 2, where L(f − g) denotes the sum of the absolute values of the coefficients of f − g. On the other hand, we show that this inequality cannot be replaced by L(f − g) ≤ 1. For this, for each integer d ≥ 15 we construct infinitely many polynomials f ∈ Z[x] of degree d such that neither f itself nor any f (x) ± x k , where k is a non-negative integer, is square-free. Polynomials over prime fields and their distances to square-free polynomials are also considered.2010 Mathematics Subject Classification. 11C08, 11T06. Key words and phrases. Integer polynomial, square-free polynomial, Turán's problem.and, moreover, at least one of them satisfieswhere f stands for the sum of the squares of the coefficients of f . In [1], Banerjee and Filaseta improved the above upper bound towhere c 0 is an effectively computable absolute constant. In addition, using computational strategies, it has been confirmed in [3,4,9,12,13] that if f ∈ Z[x] has degree d ≤ 40, then there exists an irreducible polynomial g ∈ Z[x] with deg g = d and L(f − g) ≤ 5. On the other hand, although the trivial example f (x) = x 3 shows that C ≥ 2, it is not known that the optimal constant C should be strictly greater than 2.In this paper, we consider a variant of Turán's problem, where "irreducible polynomial g" is replaced by "square-free polynomial g". For this, we pose the following conjecture: Conjecture 1.1. For any f ∈ Z[x] of degree d, there is a square-free polynomial g ∈ Z[x] of degree at most d satisfying L(f − g) ≤ 2.Another problem related to Turán's problem is that of Szegedy asking if there exists a constant C 0 depending only on d such that for any f ∈ Z[x] of degree d, the polynomial f (x) + t is irreducible for some t ∈ Z with |t| ≤ C 0 . In general, the problem of Szegedy is still open; see the papers of Győry [10] and Hajdu [11]. However, in our setting, when "irreducible" is replaced by "square-free", this problem becomes very simple. One can take, for instance, C 0 = ⌊d/2⌋.Proof. Let S be a subset of Z with the property that for each integer t ∈ S some h 2 t , where h t ∈ Z[x] is of degree at least 1, divides the polynomial f (x) + t. Then, h t = h s when t = s both belong to S, since otherwise h t | (t − s), a contradiction. Also, h t divides the derivative f ′ for every t ∈ S, so the cardinality of the set S does not exceed deg f ′ ≤ d − 1. The assertion of the theorem now follows, because the set {−⌊d/2⌋, . . . , 0, . . . , ⌊d/2⌋} contains at least d integers.

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“…For an overview of digit related results in the integers, see the recent work of Dietmann, Elsholtz and Shparlinski [2] which also contains a section on finite fields, improving an earlier result of Dartyge, Mauduit and Sárközy [1]. See also [6], which contains an extensive list of references to related problems.…”

Section: Introductionmentioning

confidence: 99%