Bernstein's Theorem (Polynomials)¶

Bernstein's polynomial theorem sharply bounds a complex polynomial's unit-circle derivative norm by its degree times its value norm.

Version: v1 · 2026-10-03 · History
Domain-specific #: 13012
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Polynomial Inequalities, Approximation Theory → Mathematics
Aliases: Bernstein's inequality for algebraic polynomials

Core Idea¶

For a complex algebraic polynomial \(P\) of degree \(n\geq1\), let \(\|P\|_{\mathbb T}=\max_{|z|=1}|P(z)|\). Bernstein's inequality states \(\|P'\|_{\mathbb T}\leq n\|P\|_{\mathbb T}\). The factor is sharp: \(P(z)=\alpha z^n\) gives equality. If \(P\) has no zeros in the open unit disk, the Erdős–Lax refinement gives \(\|P'\|_{\mathbb T}\leq(n/2)\|P\|_{\mathbb T}\); that extra hypothesis must be checked.[^{ref-864cc7c6d51d][}ref-cf8f4e61ce9f]

Scope of Application¶

This is a derivative bound for algebraic polynomials on the complex unit circle. The related trigonometric-polynomial bound has a similar form but a periodic variable and Fourier degree.[^ref-40aa07331697] The live probabilistic Bernstein inequalities, Bernstein polynomial basis, and Bernstein approximation-converse theorem concern distinct carriers and conclusions. A redirect from “Bernstein's inequality (mathematical analysis)” does not establish that the distinct inequality family is another name for this theorem.

Clarity¶

State the degree, compute the maximum of the polynomial and derivative on the same boundary, and only then ask about zero location. For \(P(z)=z^3\), the norms are 1 and 3, giving equality in \(3\cdot1\). Its interior zero rules out the \(3/2\) refinement. This is a direct constructed calculation of the source theorem's extremal family.[^ref-864cc7c6d51d]

Manages Complexity¶

The single coefficient \(n\) controls the derivative of every degree-\(n\) polynomial relative to its boundary size. For \(Q(z)=1+z^3\), roots lie on the unit circle, none inside. Its norms are 2 and 3; the sharpened ceiling is \((3/2)\cdot2=3\), attained. The general ceiling \(3\cdot2=6\) remains true but weaker. Boundary roots are allowed in the stated zero-free-open-disk condition.[^{ref-864cc7c6d51d][}ref-cf8f4e61ce9f]

Abstract Reasoning¶

The unrestricted and zero-free inequalities quantify over different classes. A stronger conclusion for a smaller class is not a contradiction, and applying \(n/2\) to \(z^3\) would fail because its premise is false. This is a logical scope distinction, not an intrinsic design tradeoff.

Knowledge Transfer¶

The useful mathematical pattern is a sharp degree-indexed norm bound plus a premise-sensitive refinement. The named theorem itself depends on algebraic-polynomial degree, complex derivative, unit-circle norm, and zero location; “complexity controls change” without those types is too vague to be a prime. The strict Formal Theorem parent does not import concentration-bound semantics. Ankeny and Rivlin's original paper restates the bound and refinement; Lax's original publisher full text was not accessible in this author check.[^{ref-864cc7c6d51d][}ref-cf8f4e61ce9f]

[^ref-864cc7c6d51d]: N. C. Ankeny and T. J. Rivlin, “On a Theorem of S. Bernstein,” Pacific Journal of Mathematics 5.6 (1955), 849–852, p.849 Theorems A and B. https://msp.org/pjm/1955/5-6/pjm-v5-n6-p01-s.pdf [^ref-cf8f4e61ce9f]: P. D. Lax, “Proof of a Conjecture of P. Erdős on the Derivative of a Polynomial,” Bulletin of the American Mathematical Society 50.8 (1944), 509–513. https://doi.org/10.1090/S0002-9904-1944-08177-9 [^ref-40aa07331697]: Hervé Queffélec and Rachid Zarouf, “On Bernstein's Inequality for Polynomials” (2019), arXiv:1903.10801. https://arxiv.org/abs/1903.10801

Relationships to Other Abstractions¶

Current abstraction Bernstein's Theorem (Polynomials) Domain-specific

Parents (1) — more general patterns this builds on

Bernstein's Theorem (Polynomials) is a kind of Formal theorem Domain-specific

Bernstein's polynomial derivative inequality is a proved formal theorem.

Hierarchy paths (2) — routes to 2 parentless roots

Bernstein's Theorem (Polynomials) → Formal theorem → Formal System → Formalization → Representation → Abstraction

Show alternative path (1)

Neighborhood in Abstraction Space¶

Bernstein's Theorem (Polynomials) sits in a moderately populated region (49^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Field Theory & Polynomial Structures (25 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08