On the geometry of random lemniscates

Lerario, Antonio; Lundberg, Erik

doi:10.1112/plms/pdw039

Cited by 13 publications

(18 citation statements)

References 55 publications

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“…1.2], the numerator p and denominator q of the rational function appear without differentiation. Based on the results in [16] we conjecture that when m = n → ∞ the average number of connected components of the random critical lemniscate (7) grows linearly (the maximum rate possible).…”

Section: 3mentioning

confidence: 97%

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On the zeros of random harmonic polynomials: The truncated model

Lerario¹,

Lundberg²

2016

Journal of Mathematical Analysis and Applications

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Abstract. Motivated by Wilmshurst's conjecture and more recent work of W. Li and A. Wei [17], we determine asymptotics for the number of zeros of random harmonic polynomials sampled from the truncated model, recently proposed by J. Hauenstein, D. Mehta, and the authors [10]. Our results confirm (and sharpen) their (3/2)−powerlaw conjecture [10] that had been formulated on the basis of computer experiments; this outcome is in contrast with that of the model studied in [17]. For the truncated model we also observe a phase-transition in the complex plane for the Kac-Rice density.

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Section: 3mentioning

confidence: 97%

“…We will apply Lebesgue's dominated convergence theorem to take the limit of the integral appearing in (16). The following claim implies that the sequence of integrands in (16) is bounded by a single integrable function.…”

Section: Proof Of Theoremmentioning

confidence: 99%

On the zeros of random harmonic polynomials: The truncated model

Lerario¹,

Lundberg²

2016

Journal of Mathematical Analysis and Applications

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“…By writing

normalΛ

as the zero set of the conjugation‐invariant polynomial

p false(z false) \bar{p (z)} - 1

, it is apparent that lemniscates are real algebraic curves. Thus, the following question fits within the general theme of studying the topology of random real algebraic varieties, a field that has seen significant recent progress .…”

Section: Introductionmentioning

confidence: 99%

“…This question was addressed in in the setting of rational lemniscates. The next theorem answers this question for a random polynomial lemniscate based on the Kac model.…”

Section: Introductionmentioning

confidence: 99%

“…It seems relevant (even if the results are rather different) to mention here the recent work of Beffara and Gayet that used methods from percolation theory to prove the existence of arbitrarily large components in zero sets of random analytic functions (namely, in the setting of a translation‐invariant Gaussian ensemble of bivariate functions from the real Bargmann–Fock space). While the proof of Theorem is simple and does not require percolation theory, the model of random rational lemniscates studied in seems to exhibit percolation of long components (according to simulations), and it would be interesting to adapt the methods from (or perhaps the more recent work ) to the setting of random rational lemniscates.…”

Section: Introductionmentioning

confidence: 99%

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The arc length and topology of a random lemniscate

Lundberg

Ramachandran

2017

J. London Math. Soc.

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A polynomial lemniscate is a curve in the complex plane defined by {z ∈ C : |p(z)| = t}. Erdös, Herzog and Piranian posed the extremal problem of determining the maximum length of a lemniscate Λ = {z ∈ C : |p(z)| = 1} when p is a monic polynomial of degree n. In this paper, we study the length and topology of a random lemniscate whose defining polynomial has independent Gaussian coefficients. When the polynomial is sampled from the Kac ensemble we show that the length approaches a nonzero constant as n → ∞. Concerning the connected components of a random lemniscate, we prove that the average number of them is asymptotically n and that there is a positive probability (independent of n) of a giant component, that is, a connected component whose length is some prescribed portion (say half) of the expected total length.

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Low-Degree Approximation of Random Polynomials

Diatta

Lerario

2021

Found Comput Math

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We prove that with “high probability” a random Kostlan polynomial in $$n+1$$ n + 1 many variables and of degree d can be approximated by a polynomial of “low degree” without changing the topology of its zero set on the sphere $$\mathbb {S}^n$$ S n . The dependence between the “low degree” of the approximation and the “high probability” is quantitative: for example, with overwhelming probability, the zero set of a Kostlan polynomial of degree d is isotopic to the zero set of a polynomial of degree $$O(\sqrt{d \log d})$$ O ( d log d ) . The proof is based on a probabilistic study of the size of $$C^1$$ C 1 -stable neighborhoods of Kostlan polynomials. As a corollary, we prove that certain topological types (e.g., curves with deep nests of ovals or hypersurfaces with rich topology) have exponentially small probability of appearing as zero sets of random Kostlan polynomials.

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On the geometry of random lemniscates

Cited by 13 publications

References 55 publications

On the zeros of random harmonic polynomials: The truncated model

On the zeros of random harmonic polynomials: The truncated model

The arc length and topology of a random lemniscate

Low-Degree Approximation of Random Polynomials

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