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arxiv: 2503.24076 · v1 · pith:NAUOFGXEnew · submitted 2025-03-31 · 🧮 math.CO

On a question about real rooted polynomials and f-polynomials of simplicial complexes

Lili Mu , Volkmar Welker This is my paper

Pith reviewed 2026-05-22 22:20 UTC · model grok-4.3

classification 🧮 math.CO
keywords real rooted polynomialsf-polynomialssimplicial complexesf-vectorsbinomial representationBell Skandera question
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The pith

Real-rootedness implies binomial representation properties that provide a sufficient criterion for a polynomial to be the f-polynomial of a simplicial complex.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The question posed by Bell and Skandera is whether real rootedness of a polynomial with positive integer coefficients implies it is the f-polynomial of some simplicial complex. The paper identifies properties that real rootedness forces on the binomial representation of the coefficients fi = binom(x_{i+1}, i+1). These properties yield a sufficient criterion guaranteeing the existence of such a complex. Additional results show that the collection of f-vectors is closed under operations preserving real rootedness, and some classical examples are confirmed to be f-polynomials.

Core claim

Real rootedness of f(t) forces the sequence x_i in the binomial representation fi = binom(x_{i+1}, i+1) to satisfy certain relations, and when these relations hold, a simplicial complex with the given f-vector exists. This gives a sufficient condition for the Bell-Skandera question. The paper also shows closure of the set of f-vectors under constructions that preserve real rootedness.

What carries the argument

The binomial representation of the coefficients fi = binom(x_{i+1}, i+1) and the conditions it must satisfy due to real rootedness of the polynomial.

If this is right

  • The sufficient criterion applies to many real-rooted polynomials, including some classical ones that are verified as f-polynomials.
  • f-vectors of simplicial complexes are closed under several constructions that preserve real rootedness of their generating polynomials.
  • Two further approaches to resolving the full question are described.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The criterion may be applicable to other families of real-rooted polynomials beyond those verified in the paper.
  • Closure under operations suggests a way to build larger examples from smaller ones while maintaining the real-rooted property.

Load-bearing premise

That the properties real rootedness imposes on the x_i sequence are enough by themselves to ensure a simplicial complex exists with the prescribed f-vector.

What would settle it

A real-rooted polynomial with positive integer coefficients whose associated x_i sequence fails the sufficient criterion, together with a proof that no simplicial complex has that f-vector.

read the original abstract

For a polynomial $f(t) = 1+f_0t+\cdots +f_{d-1}t^d$ with positive integer coefficients Bell and Skandera ask if real rootedness of f(t) implies that there is a simplicial complex with f-vector $(1,f_0 \ldots,f_{d-1})$. In this paper we discover properties implied by the real rootedness of f(t) in terms of the binomial representation $f_i = \binom{x_{i+1}}{i+1}, i \geq 0$. We use these to provide a sufficient criterion for a positive answer to the question by Bell and Skandera. We also describe two further approaches to the conjecture and use one to verify that some well studied real rooted classical polynomials are f-polynomials. Finally, we provide a series of results showing that the set of f-vectors of simplicial complexes is closed under constructions also preserving real rootedness of their generating polynomials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript addresses the Bell-Skandera question of whether real-rootedness of a polynomial f(t)=1+f_0 t+⋯+f_{d-1}t^d with positive integer coefficients implies that (1,f_0,…,f_{d-1}) is the f-vector of some simplicial complex. It derives explicit properties of the sequence (x_i) defined by the binomial representation f_i=binom(x_{i+1},i+1) that follow from real-rootedness, states a sufficient criterion on these x_i guaranteeing realizability, verifies the criterion on several classical real-rooted polynomials, and proves that the realizable set is closed under a collection of operations that also preserve real-rootedness.

Significance. The sufficient criterion and the closure theorems supply concrete, checkable conditions and constructive methods that advance the Bell-Skandera question for explicit families. The verification on classical examples and the explicit derivation of the x_i properties from real-rootedness constitute the main strengths; if the criterion is broadly applicable, the work provides a useful bridge between real-rootedness and combinatorial realizability.

minor comments (3)
  1. [Abstract] The abstract refers to 'two further approaches to the conjecture' without naming them; a brief enumeration in the introduction would help readers locate the corresponding sections.
  2. Notation for the binomial basis and the sequence x_i should be introduced with a displayed equation in §2 or §3 to avoid ambiguity when the sufficient criterion is stated.
  3. The closure results would benefit from an explicit statement of the precise operations (e.g., which products or substitutions) that preserve both real-rootedness and the f-vector property.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, significance assessment, and recommendation of minor revision. We appreciate the recognition of the sufficient criterion, closure results, and verifications as advancing the Bell-Skandera question.

Circularity Check

0 steps flagged

Derivation self-contained; no circularity detected

full rationale

The manuscript starts from the real-rootedness hypothesis on f(t), extracts explicit inequalities and monotonicity properties on the binomial sequence x_i via direct algebraic manipulation of the roots, states a sufficient criterion on those x_i that is strictly weaker than the hypothesis, and verifies the criterion on independent classical examples. Closure results are shown by explicit constructions that preserve both real-rootedness and the simplicial-complex property. No step reduces a claimed prediction or criterion back to a fitted parameter, a self-referential definition, or a load-bearing self-citation; all load-bearing implications are external to the input data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard facts about real-rooted polynomials and the definition of f-vectors; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Real-rootedness of a polynomial with positive coefficients implies specific relations when coefficients are expressed in the binomial basis.
    Invoked to obtain the sufficient criterion.

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