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On the length of lemniscates

Alexandre Eremenko, Walter Hayman

Abstract

We show that for a monic polynomial p of degree d, the length of the level set {z: |p(z)|=1} is at most 9.2 d, which improves an earlier estimate due to P. Borwein. For d=2 we show that the extremal level set is the Bernoullis' Lemniscate. One ingredient of our proofs is the fact that for an extremal polynomial this level set is connected.

On the length of lemniscates

Abstract

We show that for a monic polynomial p of degree d, the length of the level set {z: |p(z)|=1} is at most 9.2 d, which improves an earlier estimate due to P. Borwein. For d=2 we show that the extremal level set is the Bernoullis' Lemniscate. One ingredient of our proofs is the fact that for an extremal polynomial this level set is connected.

Paper Structure

This paper contains 9 theorems, 19 equations.

Key Result

Theorem 1

For monic polynomials $p$ of degree $d$$|E(p)|\leq \alpha_0 d< 9.173 d$.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7