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Monic polynomial lower bounds for the Chebyshev constant and capacity

Statement

Assume the Axiom of Choice. Let $K \subseteq C$ be nonempty and compact. Then for every monic complex polynomial $p$ of degree $n \geq 1$ ,

$∥ p ∥_{K} \geq cheb (K)^{n} and ∥ p ∥_{K} \geq cap (K)^{n},$

and consequently $cap (K) \leq cheb (K)$ .

The first lower bound and the argument at capacity zero are choice-free; the Axiom of Choice is used only through the reciprocity inequality (Reciprocity inequality for logarithmic potentials), hence through the equilibrium theory of Robin constant and logarithmic capacity of a compact set.

Facts & Assumptions

Given: A nonempty compact set $K \subseteq C$ , a monic complex polynomial $p$ of degree $n \geq 1$ , the Axiom of Choice, and the conventions of Chebyshev constant of a compact planar set, Formal complex polynomials, evaluation, degree, leading coefficient, and monic polynomials, Logarithmic potential and energy of a positive compactly supported measure and Robin constant and logarithmic capacity of a compact set.

[F1]

For nonempty compact $K$ , $∥ p ∥_{K} = \sup_{z \in K} ∣ p (z) ∣ \in [0, \infty)$ is finite and attained, and for every integer $n \geq 1$ the number $t_{n} (K) = \inf {∥ q ∥_{K} : q monic of degree n}$ is a real number with $0 \leq t_{n} (K) < \infty$ ; the Chebyshev constant is $cheb (K) = \inf_{n \geq 1} t_{n} (K)^{1 / n} \in [0, \infty)$ with nonnegative $n$ -th roots, and $cheb (\emptyset) = 0$ (Chebyshev constant of a compact planar set, Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ ).

[F2]

A monic polynomial of degree $n$ has leading coefficient $1$ ; if $p$ is monic of degree $n$ then $p$ belongs to the class whose infimum defines $t_{n} (K)$ (Formal complex polynomials, evaluation, degree, leading coefficient, and monic polynomials, Chebyshev constant of a compact planar set).

[F3]

A polynomial $f$ of degree $n \geq 1$ factors as $f (x) = c \prod_{j = 1}^{r} (x - α_{j})^{m_{j}}$ with distinct roots $α_{j}$ , positive multiplicities $m_{j}$ summing to $n$ , and $c$ its leading coefficient; equivalently $f$ has exactly $n$ roots counted with multiplicity (A complex polynomial of degree $n$ has exactly $n$ roots counted with multiplicity).

[F4]

For points $α_{1}, \dots, α_{n} \in C$ the Dirac measures $δ_{α_{j}}$ are Borel probability measures (A Dirac set function is a probability measure, The Dirac set function at a point), finite nonnegative weighted sums of measures are measures (Nonnegative scalar multiples and countable weighted sums of measures are measures), and $ν = \frac{1}{n} \sum_{j = 1}^{n} δ_{α_{j}}$ is thus a Borel probability measure carried by the finite set ${α_{1}, \dots, α_{n}}$ , which is compact (Probability measures and probability spaces).

[F5]

For a finite positive Borel measure $ν$ of compact support, $U^{ν} (z) = \int_{C} k (z, w) d ν (w)$ with $k (z, w) = \log \frac{1}{∣ z - w ∣}$ and diagonal value $+ \infty$ (Logarithmic potential and energy of a positive compactly supported measure).

[F6]

For nonempty compact $F$ , $V_{F} = \inf_{μ \in P (F)} I (μ) \in (- \infty, + \infty]$ and $cap (F) = \exp (- V_{F})$ when $V_{F} < + \infty$ , while $cap (F) = 0$ when $V_{F} = + \infty$ ; in particular $cap (K) > 0$ is equivalent to $V_{K} < + \infty$ , and then $cap (K) = \exp (- V_{K})$ (Robin constant and logarithmic capacity of a compact set).

[F7]

Assume the Axiom of Choice. If $K$ is compact with $cap (K) > 0$ and $σ$ is any compactly supported Borel probability measure on $C$ , then $\inf_{z \in K} U^{σ} (z) \leq V_{K} = \log \frac{1}{cap (K)}$ (Reciprocity inequality for logarithmic potentials).

[F8]

The modulus is multiplicative: $∣ \prod_{j} w_{j} ∣ = \prod_{j} ∣ w_{j} ∣$ for finitely many complex numbers, and $∣ w ∣ = 0$ exactly when $w = 0$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

Proof

technique · direct

1.1F1F2given

By [F1] and [F2] the number $t_{n} (K)$ is a well-defined real number with $t_{n} (K) \leq ∥ p ∥_{K} < + \infty$ , since $p$ is monic of degree $n$ and all $∥ q ∥_{K}$ are nonnegative; and $cheb (K) = \inf_{m \geq 1} t_{m} (K)^{1 / m}$ is a nonnegative real number, so $cheb (K) \leq t_{n} (K)^{1 / n}$ .

2.1step 1.1F1algebra

First lower bound. Since raising preserves the order on nonnegative reals, $cheb (K) \leq t_{n} (K)^{1 / n}$ from step 1.1 gives $cheb (K)^{n} \leq t_{n} (K)$ ; and $t_{n} (K) \leq ∥ p ∥_{K}$ because $p$ belongs to the class whose infimum is $t_{n} (K)$ ; hence $∥ p ∥_{K} \geq cheb (K)^{n}$ , the first asserted inequality.

2.2step 1.1F1algebra

Second bound when $cap (K) = 0$ . If $cap (K) = 0$ then $cap (K)^{n} = 0$ while $∥ p ∥_{K} \geq 0$ , so $∥ p ∥_{K} \geq cap (K)^{n}$ holds trivially.

2.3step 1.1F2F3F4F5F6F7F8algebra

Second bound when $cap (K) > 0$ . By [F3], applied to $p$ of degree $n$ with leading coefficient $1$ (step 1.1 and [F2]), there are distinct roots $α_{1}, \dots, α_{r}$ with multiplicities $m_{1}, \dots, m_{r}$ summing to $n$ and $p (z) = \prod_{j = 1}^{r} (z - α_{j})^{m_{j}}$ ; listing the roots with multiplicity as $α_{1}, \dots, α_{n}$ and putting $ν : = \frac{1}{n} \sum_{j = 1}^{n} δ_{α_{j}}$ , [F4] makes $ν$ a Borel probability measure whose finite support is compact. By [F5] and [F8], $U^{ν} (z) = \frac{1}{n} \sum_{j = 1}^{n} \log \frac{1}{∣ z - α_{j} ∣} = \frac{1}{n} \log \frac{1}{∣ p (z) ∣}$ for every $z \in C$ , both sides being $+ \infty$ exactly at the roots of $p$ . By [F7], whose Axiom of Choice hypothesis is part of the Given, $\inf_{z \in K} U^{ν} (z) \leq V_{K} < + \infty$ ; so for each real $ε > 0$ there is $z_{ε} \in K$ with $U^{ν} (z_{ε}) \leq V_{K} + ε$ , and then $p (z_{ε}) \neq 0$ and $\log \frac{1}{∣ p (z_{ε}) ∣} = n U^{ν} (z_{ε}) \leq n (V_{K} + ε)$ , that is, $∣ p (z_{ε}) ∣ \geq e^{- n (V_{K} + ε)}$ . Hence $∥ p ∥_{K} \geq e^{- n (V_{K} + ε)}$ for every $ε > 0$ , and letting $ε ↓ 0$ gives $∥ p ∥_{K} \geq e^{- n V_{K}} = cap (K)^{n}$ by [F6] and algebra.

3.1step 2.2step 2.3F1algebra

Capacity is at most the Chebyshev constant. If $cap (K) = 0$ then $cap (K) = 0 \leq cheb (K)$ by the nonnegativity in [F1]. If $cap (K) > 0$ , step 2.3 gives $∥ q ∥_{K} \geq cap (K)^{n}$ for every monic $q$ of degree $n$ , so the infimum satisfies $t_{n} (K) \geq cap (K)^{n} > 0$ and, taking nonnegative $n$ -th roots, $t_{n} (K)^{1 / n} \geq cap (K)$ for every $n \geq 1$ ; since $cheb (K)$ is the infimum of these numbers, $cheb (K) \geq cap (K)$ . In either case $cap (K) \leq cheb (K)$ , the final assertion.

4.1step 2.1step 2.2step 2.3step 3.1F7∎

Assembly. Step 2.1 gives the lower bound by $cheb (K)^{n}$ and steps 2.2 and 2.3 together give the lower bound by $cap (K)^{n}$ for every monic $p$ of degree $n \geq 1$ ; step 3.1 gives $cap (K) \leq cheb (K)$ . The Axiom of Choice is used only in [F7]; the factorisation, the Dirac measures, the weighted sum and all estimates are choice-free.

Remarks

What is used from the equilibrium theory. The second bound is the point where the logarithmic potential of the zero-counting measure of $p$ meets the capacity: the reciprocity inequality (Reciprocity inequality for logarithmic potentials) says that the potential of any compactly supported probability measure, including one concentrated on the roots of $p$ outside $K$ , dips to at most $V_{K}$ somewhere on $K$ . When $cap (K) = 0$ no equilibrium measure exists and the bound degenerates to the trivial $∥ p ∥_{K} \geq 0$ .

Strictness is not asserted. The lemma only produces lower bounds; equality $cap (K) = cheb (K)$ is the content of Fekete–Szegő equality of logarithmic capacity, transfinite diameter, and Chebyshev constant, which combines the converse inequality obtained from Fekete points with the present lemma. No uniqueness of an extremal monic polynomial is claimed here.

Depends on

Used by

Fekete–Szegő equality of logarithmic capacity, transfinite diameter, and Chebyshev constant Theorem

Dependency tree · two levels

56 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. B. Saff, Logarithmic Potential Theory with Applications to Approximation Theory, §1 (standard reference, not scraped)