TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.
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The exact polynomial degree of OR is N

Statement

For $N \geq 1$ , let ${O R}_{N} (x) = 0$ when $x = 0^{N}$ and $1$ otherwise. Its exact approximate degree is ${\overset{}{\deg}}_{0} ({O R}_{N}; {0, 1}^{N}) = N .$

Facts & Assumptions

Given: An integer $N \geq 1$ and the total Boolean function ${O R}_{N}$ .

[F1]

Acceptance polynomials and approximate degree on Boolean domains: exact approximate degree is the least total-degree bound of a real polynomial agreeing with the function on the domain.

[F2]

Symmetrizing a Boolean-cube polynomial gives a Hamming-weight polynomial: a permutation-invariant exact approximant can be replaced by a univariate polynomial $q (t)$ of no larger degree taking the same values on every Hamming-weight layer.

[F3]

A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots: a nonzero degree- $m$ real polynomial has at most $m$ distinct real roots.

[F4]

Polynomial rings in finitely many commuting indeterminates by iteration and Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ : finite products of linear coordinate polynomials are real polynomials with total degree at most the number of factors.

Proof

1.1givenF1F4algebra

The polynomial $P (X) = 1 - \prod_{i = 0}^{N - 1} (1 - X_{i})$ has total degree at most $N$ by [F4]. At a Boolean vector $x$ , the product is $1$ exactly when all bits are zero and is $0$ otherwise, so $P (x) = {O R}_{N} (x)$ . Thus [F1] gives ${\overset{}{\deg}}_{0} ({O R}_{N}) \leq N$ .

1.2givenF2F3algebra

Let $p$ be any exact polynomial approximant of degree at most $d$ . Since OR is invariant under coordinate permutations, [F2] gives a univariate $q$ of degree at most $d$ with $q (0) = 0$ and $q (k) = 1$ for each integer $1 \leq k \leq N$ . Therefore the nonzero polynomial $q - 1$ has the $N$ distinct roots $1, \dots, N$ and is nonzero because $(q - 1) (0) = - 1$ . By [F3], its degree is at least $N$ , so $d \geq N$ . Hence every exact approximant has degree at least $N$ .

2.1step 1.1step 1.2∎

The upper and lower bounds of steps 1.1 and 1.2 coincide, proving the equality.

Depends on

Used by

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Sources

Andrew M. Childs, Lecture Notes on Quantum Algorithms (2025), §21.6 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes (arXiv:1907.09415), §11.2 (standard reference, not scraped)