Problem 1114

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claims/: The 1 claim page of Problem 1114, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f(x)\in \mathbb{R}[x]$ be a polynomial of degree $n$ whose roots $\{a_0<\cdots<a_n\}$ are all real and form an arithmetic progression.

The differences between consecutive zeros of $f'(x)$, beginning from the midpoint of $(a_0,a_m)$ towards the endpoints, are monotonically increasing.

Statement (corrected). Let $f(x)\in \mathbb{R}[x]$ be a polynomial of degree $n+1$ whose roots $\{a_0<\cdots<a_n\}$ are all real and form an arithmetic progression.

The differences between consecutive zeros of $f'(x)$, beginning from the midpoint of $(a_0,a_n)$ towards the endpoints, are monotonically increasing.

Notes. The site's wording fails for every $n$: a polynomial of degree $n$ has at most $n$ zeros, so none has the $n+1$ distinct roots $a_0<\cdots<a_n$ and the statement holds only vacuously; and the index $m$ in $(a_0,a_m)$ is never introduced. The change replaces "degree $n$" with "degree $n+1$" and "$(a_0,a_m)$" with "$(a_0,a_n)$"; nothing else changes. The site prints no source for the conjecture. Bálint [Ba60b] states it as Erdős's in his opening paragraph (p. 33), apart from his proof, for $q(y)=\prod_{m=0}^n(y-a_m)$, a polynomial of degree $n+1$ with $a_m-a_{m-1}=d$ for $m=1,\ldots,n$, and the midpoint $(a_0+a_n)/2$ of the interval $(a_0,a_n)$; his Russian summary (p. 39) states the same. His English summary (p. 40) writes that midpoint as $(a_0+a_m)/2$ of $(a_0,a_m)$, reusing the product's index, the form the site's stray index follows. Lorch [Lo76] (p. 293), independent of Bálint's proof, states Erdős's conjecture for an arbitrary polynomial with simple, equally spaced, real zeros and measures the gaps outward from the center of symmetry of its zeros, so the degree equals the number of zeros and the interval runs from the first zero to the last. The failure is not a boundary failure, since it holds at every $n$; the change rests on the setting these sources state. No text of Erdős's on the question is known, and the site's commentary records that Bálint gives none; the degree slip is the site's, and the stray index follows Bálint's English summary. No result concerns the site's wording.

Status. Proved, the site's label PROVED. Bálint proved the corrected Statement in 1960 (published in Matematikai Lapok; the accepted claim page is Bálint 1960), and Lorch gave generalizations in 1976. No Lean proof that the corpus built and audited checks the statement.

Source. erdosproblems.com/1114, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1114, https://www.erdosproblems.com/1114.

References.

• [Ba60b] Bálint, Elemér, Proof of a conjecture of P. Erdős. Mat. Lapok (1960), 33-40.
• [Lo76] Lorch, L., Some monotonicity properties of polynomials with equally spaced zeros. Acta Math. Acad. Sci. Hungar. 27 (1976), 293-300.

Formalization. No formal-conjectures statement; the claim page for Bálint 1960 links a Lean 4 proof in the lean-proofs repository at its pinned commit and records the corpus's coverage of it.

Current assessment

The question (site formulation). That for a real polynomial whose zeros $a_0<\cdots<a_n$ are real and form an arithmetic progression, the gaps between consecutive zeros of the derivative increase from the midpoint of the zeros toward either endpoint. PROVED. The corrected Statement gives the degree as $n+1$ and the interval as $(a_0,a_n)$, as Bálint states the conjecture (Notes above), and Bálint proves it. The derivative has one simple zero in each gap between consecutive zeros by Rolle's theorem, as the site's commentary (page last edited 2025-12-29) notes, and the commentary records that Bálint gives no source for the conjecture, presumably Erdős in personal communication.

Standing. One accepted full claim, Bálint 1960, published in Matematikai Lapok 11 (1960), 33–40, credited by the site's curator and cited by Lorch [Lo76] as the verification of Erdős's conjecture. The theorem statement and the shape of the proof were checked against the paper; the proof is not compiled in this wiki.

Lorch's generalizations. Lorch [Lo76] (card) proves, for the normalized polynomials with zeros at consecutive integers, that the areas and maxima of consecutive arches and the slopes at the zeros increase outward, that adjoining a zero moves the derivative's zeros toward the center of symmetry, and that the gaps between consecutive derivative zeros tend to $1$ as the degree grows. His Section 3 gives a new proof of Bálint's intermediate inequality, that consecutive derivative zeros in the normalized coordinates differ by more than one, but cites and does not reprove the monotonicity theorem itself, so the paper carries no claim page. Whether the convergence of the gaps is monotone in the degree is left open there.

Formalization. The site labels the problem PROVED, not PROVED (LEAN), and no formal-conjectures statement file exists for the problem. A Lean 4 file in Boris Alexeev's lean-proofs repository declares itself a formalization of Bálint's solution and proves the gap monotonicity for a polynomial of degree $N+1$ with $N+1$ zeros in arithmetic progression, the corrected Statement; the claim page links it at its pinned commit. The corpus has not built or audited it, so no formalized evidence is listed.

Search scope and coverage. Search scope: the site's problem page and its empty thread, the formal-conjectures tree and the lean-proofs file, as of 2026-10-07. Coverage: the statements of Bálint 1960 and Lorch 1976, Bálint's English summary and Lorch's account of Bálint's result were checked against the papers; the proofs are not compiled in this wiki.

Progress

Bálint's theorem [Ba60b] settles the question; Lorch [Lo76] adds further monotonicity properties of the same polynomials. The proofs are not compiled in this wiki.

Known Results

• [Ba60b], Bálint, main theorem: if a polynomial has only real zeros $a_0<\cdots<a_n$ in arithmetic progression, the gaps between consecutive zeros of its derivative increase from the midpoint $(a_0+a_n)/2$ toward the endpoints; after the affine normalization to zeros $0,1,\ldots,n$ this is $t_{k+1}-t_k>t_k-t_{k-1}$ for the derivative zeros $t_k\in(k-1,k)$ on the outward side, with the zeros symmetric about $n/2$.
• [Lo76], Lorch, statements (I) and (II) and Sections 3 and 4: for the normalized polynomials with zeros at consecutive integers, arch areas, arch maxima and slopes at the zeros increase outward, the derivative's zeros move toward the center when a zero is adjoined, and consecutive derivative-zero gaps tend to $1$ with the degree.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• balint_1960_proof_conjecture_erdos
• balint_1960_proof_conjecture_erdos / main_theorem
• balint_1960_proof_conjecture_erdos / statement_ii
• lorch_1976_monotonicity_properties_polynomials_equally_spaced_zeros
• lorch_1976_monotonicity_properties_polynomials_equally_spaced_zeros / conjecture_21
• lorch_1976_monotonicity_properties_polynomials_equally_spaced_zeros / equation_13
• lorch_1976_monotonicity_properties_polynomials_equally_spaced_zeros / equation_14
• lorch_1976_monotonicity_properties_polynomials_equally_spaced_zeros / equations_7_11
• lorch_1976_monotonicity_properties_polynomials_equally_spaced_zeros / statement_i