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The large-circle loop of $z^{3} - 2 z + 2$ on $∣ z ∣ = 5$ has degree three

Example

For $p (z) = z^{3} - 2 z + 2$ and $R = 5$ , the normalized based loop obtained from $p (5 h (u))$ has degree $3$ . A coefficient-scaling homotopy is

$p_{s} (z) = z^{3} + s (- 2 z + 2), 0 \leq s \leq 1.$

Facts & Assumptions

Given: The complex polynomial $p (z) = z^{3} - 2 z + 2$ and the radius $R = 5$ .

[F1]

For a nonzero polynomial, its degree is its final coefficient index, and it is monic when its leading coefficient is $1$ (Formal complex polynomials, evaluation, degree, leading coefficient, and monic polynomials).

[L1]

For a monic polynomial of positive degree $n$ , a radius greater than $1$ and the sum of the moduli of all lower coefficients gives a normalized circle loop of degree $n$ ; coefficient scaling supplies the homotopy to the standard $n$ -fold loop (The normalized large-radius loop of a monic degree- $n$ polynomial has degree $n$ ).

Verification

technique · direct

1.1givenF1algebra

The coefficient list is $(2, - 2, 0, 1)$ , so $p$ is monic of degree $3$ and the lower-coefficient modulus sum is $2 + 2 + 0 = 4 < 5$ .

2.1step 1.1L1algebra∎

The hypotheses of [L1] hold with $n = 3$ and $R = 5$ , so the normalized loop has degree $3$ . Explicitly, on $∣ z ∣ = 5$ and for $0 \leq s \leq 1$ , one has $∣ s (- 2 z + 2) ∣ \leq s (10 + 2) \leq 12 < 125 = ∣ z^{3} ∣$ , so the displayed homotopy never meets zero; its endpoints are $z^{3}$ and $z^{3} - 2 z + 2$ .

Depends on

Used by

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Dependency tree · two levels

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Sources

Allen Hatcher, Algebraic Topology, proof of Theorem 1.8 (standard reference, not scraped)

The large-circle loop of z3−2z+2z3−2z+2 on ∣z∣=5∣z∣=5 has degree three