Krishnapur 2025 area polynomial lemniscates

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lemma_9: States that for every t > 0 and every polynomial p of degree n, monic or not, the inradius of the t-level lemniscate is at least 1/(72 pi sqrt(pi)) times the square root of its area divided by n, confirming a 2009 conjecture of Solynin and Williams on the Cuenya–Levis constant.

theorem_1: States that for all large n the minimal area of the level-1 lemniscate with zeros in the closed unit disc is at most the circle version and at least a third of the circle version in degree n(log n)^4, with both bounded by c/log n from below and C/log log n from above.

theorem_2: States that for each level t > 1 there are constants depending only on t such that for all large n the minimal area of the t-level lemniscate, with zeros in the closed disc or on the circle, lies between c/log log n and C/log log n.

theorem_3: States that for each level t in (0,1) there are constants depending only on t with c/n^4 <= kappa_n(closed disc,t) <= kappa_n(T,t) <= C/n for all n >= 1, and that kappa_n(T,t) >= c/(n^2 log n) for zeros on the circle.

theorem_6: States that if K is the closure of a bounded open set with C^2-smooth boundary and K has logarithmic capacity 1, then the infimum over n of the minimal area of {|p| <= 1}, over monic degree-n polynomials with all zeros in K, is 0.

theorem_7: States that if K is compact with capacity 1, t > 0 is fixed and monic p_n with zeros in K satisfy m(Lambda_{p_n}(t) ∩ K) -> 0, then the empirical measures of the zeros of p_n converge weakly to the equilibrium measure of K.

theorem_8: States that the infimum of the inradius of the level-1 lemniscate, over monic degree-n polynomials with all zeros in the closed unit disc, is at least c/(n sqrt(log n)), improving Pommerenke's c/n^2.

theorem_p2: States the paper's main theorem: for n >= 3 the infimum of the area of {|p| < 1}, over monic polynomials of degree n with all zeros in the closed unit disc, is at least c/log n and at most C/log log n.

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Manjunath Krishnapur, Erik Lundberg, Koushik Ramachandran, On the area of polynomial lemniscates. arXiv:2503.18270 (2025). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2503.18270), every other right reserved.

For monic degree-n polynomials with all zeros in the closed unit disc, the authors prove c/log n <= inf_p m({|p| < 1}) <= C/log log n for n >= 3 (the unnumbered Theorem, p. 2), improving Pommerenke's 1961 lower bound of order n^{-4} and Wagner's 1988 upper bound of order (log log n)^{-1/2+δ}; this follows from Theorem 1, which compares the closed-disc constraint with zeros on the unit circle at degree n(log n)^4. They also determine the sharp order (log log n)^{-1} of the minimal area of {|p| <= t} for each level t > 1 (Theorem 2) and prove the bounds c/n^4 and C/n for 0 < t < 1, with c/(n^2 log n) for zeros on the circle (Theorem 3). For the inradius they prove a quantitative form of the Cuenya–Levis inequality with constant of order 1/n, confirming the Solynin–Williams conjecture (Lemma 9), and deduce the lower bound c/(n sqrt(log n)) (Theorem 8), against the conjectured 1/n of Erdős, Herzog and Piranian. For a compact set K of unit capacity, polynomials with zeros in K whose t-level lemniscates meet K in area tending to zero have zero-counting measures converging weakly to the equilibrium measure of K (Theorem 7), and inf_n κ_n(K, 1) = 0 when K is the closure of a bounded open set with C^2 boundary and capacity 1 (Theorem 6). The abstract describes Theorem 6 as showing that the minimal area converges to zero, an affirmative answer to another Erdős–Herzog–Piranian question; the theorem as printed gives the infimum over n, not a limit.

Source: https://arxiv.org/abs/2503.18270; the edition read is arXiv:2503.18270v1 (24 March 2025), whose labels and pages the result pages cite.

Bears on.

• #116: the lower bound c/log n of the Theorem on p. 2 is the problem's parenthetical (log n)^{-O(1)} form with exponent 1, which the paper states as Erdős's (log n)^{-1} question.
• #1039: Theorem 8 gives ρ(f) >= c/(n sqrt(log n)) for every admissible f, short of the problem's 1/n.
• #1040: Theorem 6 gives μ(K) = 0 for closures of bounded open sets with C^2-smooth boundary and transfinite diameter 1, a case of the problem's second question.

Results.

• Theorem (p. 2): for n >= 3, c/log n <= inf m({|p| < 1}) <= C/log log n over monic degree-n p with all zeros in the closed unit disc.
• Theorem 1 (p. 5): for all large n, c/log n <= (1/3) κ_{n(log n)^4}(T, 1) <= κ_n(closed disc, 1) <= κ_n(T, 1) <= C/log log n.
• Theorem 2 (p. 5): for t > 1, the same chain with (log log n)^4 in the degree, between c/log log n and C/log log n, constants depending only on t.
• Theorem 3 (p. 5): for t in (0, 1) and all n >= 1, c/n^4 <= κ_n(closed disc, t) <= κ_n(T, t) <= C/n, and κ_n(T, t) >= c/(n^2 log n).
• Theorem 6 (p. 5): inf_n κ_n(K, 1) = 0 for K the closure of a bounded open set with C^2-smooth boundary and capacity 1.
• Theorem 7 (p. 6): for K compact of capacity 1 and fixed t > 0, if p_n in P_n(K) have m(Λ_{p_n}(t) ∩ K) -> 0, the empirical zero measures converge weakly to the equilibrium measure of K.
• Theorem 8 (p. 6): the minimal inradius ρ_n over P_n(closed disc) is at least c/(n sqrt(log n)).
• Lemma 9 (p. 6): for t > 0 and any degree-n polynomial p, monic or not, ρ(Λ_p(t)) >= sqrt(m(Λ_p(t))) / (72 π sqrt(π) n).

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