Some recent results on the geometry of complex polynomials: the Gauss–Lucas theorem, polynomial lemniscates, shape analysis, and conformal equivalence

Richards, Trevor

doi:10.1007/s40627-021-00079-8

Cited by 2 publications

(1 citation statement)

References 22 publications

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“…Lemniscates also have appeared prominently in the theory and application of conformal mapping [3], [15], [13]. See also the recent survey [36] which elaborates on some of the more recent of the above mentioned lines of research. 1.3.…”

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confidence: 99%

Inradius of random lemniscates

Krishnapur¹,

Lundberg²,

Ramachandran³

2023

Preprint

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A classically studied geometric property associated to a complex polynomial p is the inradius (the radius of the largest inscribed disk) of its (filled) lemniscate Λ := {z ∈ C : |p(z)| < 1}.In this paper, we study the lemniscate inradius when the defining polynomial p is random, namely, with the zeros of p sampled independently from a compactly supported probability measure µ. If the negative set of the logarithmic potential Uµ generated by µ is non-empty, then the inradius is bounded from below by a positive constant with overwhelming probability. Moreover, the inradius has a determinstic limit if the negative set of Uµ additionally contains the support of µ.On the other hand, when the zeros are sampled independently and uniformly from the unit circle, then the inradius converges in distribution to a random variable taking values in (0, 1/2).We also consider the characteristic polynomial of a Ginibre random matrix whose lemniscate we show is close to the unit disk with overwhelming probability.

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confidence: 99%