LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A root-free complex polynomial gives nullhomotopic normalized circle loops

Statement

Let $p$ be a complex polynomial with no zero in $C$ . For each real $R \geq 0$ , evaluate $p$ on the circle of radius $R$ , divide by its value at the basepoint $R$ , and radially normalize to the unit circle. Transported through the homeomorphism $h : R / Z \to S^{1}$ , this is a based circle loop $α_{R}$ , and every $α_{R}$ is nullhomotopic. In particular, every such loop has degree zero.

Facts & Assumptions

Given: A complex polynomial $p$ such that $p (z) \neq 0$ for every $z \in C$ , a real $R \geq 0$ , and the unit-circle homeomorphism $h : R / Z \to S^{1}$ .

[F1]

A complex polynomial is a finite coefficient list, and its evaluation at $z \in C$ is the corresponding finite sum of powers of $z$ (Formal complex polynomials, evaluation, degree, leading coefficient, and monic polynomials).

[F2]

Under $C = R^{2}$ , complex continuity is continuity for the Euclidean metric (The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane).

[L1]

Complex modulus satisfies $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ , $∣ z ∣ = 0$ exactly when $z = 0$ , and $∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[L2]

The map $h : R / Z \to S^{1}$ , $h ([t]) = (\cos 2 π t, \sin 2 π t)$ , is a homeomorphism and sends $[0]$ to $1 \in S^{1}$ ( $[t] \mapsto (\cos 2 π t, \sin 2 π t)$ is a homeomorphism from $R / Z$ to the unit circle).

[L3]

A map into a finite product is continuous exactly when each component is continuous, and continuity of maps into $R^{2}$ is componentwise (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice, A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions).

[L4]

Finite sums and products of continuous real-valued maps are continuous, as are quotients on cozero sets (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined).

[L5]

A based circle loop is nullhomotopic exactly when its degree is zero (A based circle loop is nullhomotopic exactly when its degree is zero).

Proof

technique · constructive

1.1

For $τ \in [0, 1]$ and $u \in R / Z$ , put $q (τ, u) = p (τ R h (u)) / p (τ R)$ . Root-freeness makes numerator and denominator nonzero, so $A (τ, u) : = h^{- 1} (\frac{q (τ, u)}{∣ q (τ, u) ∣})$ is defined; moreover $q (τ, [0]) = 1$ , hence $A (τ, [0]) = [0]$ . Define $α_{R} (u) = A (1, u)$ .

givenF1L1L2construct

2.1

Writing complex addition and multiplication in real and imaginary coordinates shows from [F1], [L3], and [L4] that $(τ, u) \mapsto p (τ R h (u))$ and $τ \mapsto p (τ R)$ are continuous. Root-freeness and [L1] make division and radial normalization continuous, and [L2] makes $A$ continuous. At $τ = 0$ one has $q (0, u) = p (0) / p (0) = 1$ , so $A (0, u) = [0]$ for every $u$ , while step 1.1 keeps the basepoint fixed for all $τ$ . Thus $A$ is a based homotopy on the unit interval from the constant loop to $α_{R}$ , including the case $R = 0$ without division by $R$ .

step 1.1F1F2L1L2L3L4

3.1

Hence $α_{R}$ is nullhomotopic for every $R \geq 0$ , and [L5] gives $\deg (α_{R}) = 0$ .

step 2.1L5discharge-construct∎

Depends on

Used by

Fundamental theorem of algebra by the fundamental-group obstruction Theorem

Dependency tree · two levels

71 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

Allen Hatcher, Algebraic Topology, proof of Theorem 1.8 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Chapter 1, §7 (standard reference, not scraped)