Irreducibility of random polynomials: general measures

Bary-Soroker, Lior; Koukoulopoulos, Dimitris; Kozma, Gady

doi:10.48550/arxiv.2007.14567

Cited by 4 publications

(24 citation statements)

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“…, L} for L divisible by at least 4 distinct primes, was established (unconditionally) by Bary-Soroker and Kozma [2]. In recent work, Bary-Soroker, Koukoulopoulos, and Kozma [1] showed that the result continues to hold for {1, . .…”

Section: Introductionmentioning

confidence: 91%

“…Rademacher random variables. In [8], Eberhard asked if ϕ(t) is irreducible with probability 1− o n (1). We answer this question in the affirmative under the extended Riemann hypothesis.…”

Section: Introductionmentioning

confidence: 91%

“…Ber(1/2) random variables (i.e. each b i is independently 0 or 1 with probability 1/2 each), is irreducible in Z[x] with probability 1−o d (1). This was established in a more general form by Breuillard and Varjú [3] under the Riemann Hypothesis for a family of Dedekind zeta functions.…”

Section: Introductionmentioning

confidence: 93%

“…, L} for L ≥ 35. We refer the reader to [1][2][3] for more precise and general versions of the aforementioned results.…”

Section: Introductionmentioning

confidence: 99%

“…It was noted in [8] that given the techniques in [3,8], Theorem 1.1 (with the weaker probability bound 1 − o n (1)) can be deduced from the following universality statement: the probability that tI n − M n is invertible over F p , for p = n Ω (1) , is essentially the same as for an n × n symmetric matrix whose entries on and above the diagonal are sampled from the uniform distribution on F p . Despite the intensive efforts to study the singularity probability of symmetric Rademacher matrices ([4,6,7, 10,14,20,24] and especially the recent breakthrough [5] which confirms the long-standing conjecture that the singularity probability of symmetric Rademacher matrices is exponentially small), a result of this precision has remained elusive.…”

Section: Introductionmentioning

confidence: 99%

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Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

Ferber¹,

Jain²,

Sah³

et al. 2021

Preprint

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Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric {±1}-matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random {±1}-matrices over Fp for primes 2 < p ≤ exp(O(n 1/4 )). Previously, such estimates were available only for p = o(n 1/8 ). At the heart of our proof is a way to combine multiple inverse Littlewood-Offord-type results to control the contribution to singularity-type events of vectors in F n p with anticoncentration at least 1/p + Ω(1/p 2 ). Previously, inverse Littlewood-Offord-type results only allowed control over vectors with anticoncentration at least C/p for some large constant C > 1.and applying Theorem B.2 once again finishes.

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 93%

“…, L} for L ≥ 35. We refer the reader to [1][2][3] for more precise and general versions of the aforementioned results.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

Ferber¹,

Jain²,

Sah³

et al. 2021

Preprint

View full text Add to dashboard Cite

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Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

Ferber¹,

Jain²,

Sah³

et al. 2022

Math. Proc. Camb. Phil. Soc.

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Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$ -matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random $\{\pm 1\}$ -matrices over $\mathbb{F}_p$ for primes $2 < p \leq \exp(O(n^{1/4}))$ . Previously, such estimates were available only for $p = o(n^{1/8})$ . At the heart of our proof is a way to combine multiple inverse Littlewood–Offord-type results to control the contribution to singularity-type events of vectors in $\mathbb{F}_p^{n}$ with anticoncentration at least $1/p + \Omega(1/p^2)$ . Previously, inverse Littlewood–Offord-type results only allowed control over vectors with anticoncentration at least $C/p$ for some large constant $C > 1$ .

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Towards van der Waerden’s conjecture

Chow

Dietmann

2023

Trans. Amer. Math. Soc.

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How often is a quintic polynomial solvable by radicals? We establish that the number of such polynomials, monic and irreducible with integer coefficients in [ − H , H ] [-H,H] , is O ( H 3.91 ) O(H^{3.91}) . More generally, we show that if n ⩾ 3 n \geqslant 3 and n ∉ { 7 , 8 , 10 } n \notin \{ 7, 8, 10 \} then there are O ( H n − 1.017 ) O(H^{n-1.017}) monic, irreducible polynomials of degree n n with integer coefficients in [ − H , H ] [-H,H] and Galois group not containing A n A_n . Save for the alternating group and degrees 7 , 8 , 10 7,8,10 , this establishes a 1936 conjecture of van der Waerden.

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Irreducibility of random polynomials: general measures

Cited by 4 publications

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Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

Towards van der Waerden’s conjecture

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