Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern

Feihu Liu; Guoce Xin; Sihao Tao

arxiv: 2605.23544 · v2 · pith:UE6TGDHKnew · submitted 2026-05-22 · 🧮 math.CO

Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern

Feihu Liu , Sihao Tao , Guoce Xin This is my paper

Pith reviewed 2026-05-25 04:27 UTC · model grok-4.3

classification 🧮 math.CO

keywords Ehrhart polynomialsign patternintegral polytopeCartesian productReeve tetrahedronsimplexh-star polynomial

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The pith

Cartesian products of adjustable simplices and the Reeve tetrahedron realize any sign pattern in the Ehrhart polynomial of a d-dimensional polytope.

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

The paper shows that for any dimension d at least 3 and any choice of positions between 1 and d-2, there exists an integral polytope whose Ehrhart polynomial has negative coefficients exactly at those positions and positive coefficients everywhere else. The construction begins with a family of simplices S_d(m) whose Ehrhart polynomials have leading coefficient m and all other coefficients fixed positive constants, then forms the Cartesian product with the Reeve tetrahedron. This product allows the parameter m to be chosen large enough to set the desired signs independently. A sympathetic reader would care because the sign pattern problem asks what coefficient behaviors are geometrically possible, and a complete affirmative answer clarifies the range of Ehrhart polynomials that can arise from actual polytopes.

Core claim

The authors construct simplices S_d(m) whose Ehrhart polynomials take the form m t^d plus fixed positive lower-degree terms. They then form the Cartesian product of S_d(m) with the Reeve tetrahedron and prove that, for sufficiently large m, the resulting Ehrhart polynomial has negative coefficients precisely in any prescribed degrees between 1 and d-2 while keeping all other coefficients positive. This yields an explicit polytope for every sign pattern and every d at least 3, giving a complete solution to the problem.

What carries the argument

The Cartesian product of the simplices S_d(m) and the Reeve tetrahedron, whose Ehrhart polynomial is the product of the individual polynomials and thereby separates the sign control via the free parameter m.

If this is right

Every combination of negative coefficients in degrees 1 through d-2 is attained by some integral polytope.
The sign pattern problem has an affirmative answer in every dimension d at least 3.
A fast algorithm computes the h-star polynomial for the simplices Delta(0,q).

Where Pith is reading between the lines

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

The same product technique might be used to prescribe signs in other polynomial invariants attached to polytopes.
No further sign restrictions on Ehrhart polynomials exist beyond the known positivity of the leading coefficient and constant term.
Explicit families like S_d(m) could be combined with other base polytopes to control additional Ehrhart features.

Load-bearing premise

The Ehrhart polynomial of the Cartesian product factors so that raising m flips the signs of the chosen coefficients without flipping the signs of the others.

What would settle it

A direct computation of the coefficients in the product polynomial for some choice of indices and some large m showing that at least one targeted coefficient remains positive or an untargeted coefficient becomes negative.

read the original abstract

In Ehrhart theory, the well-known sign pattern problem asks: given a positive integer $d\geq 3$ and integers $1 \leq i_1 < \cdots < i_k \leq d-2$, does there exist a $d$-dimensional integral polytope $\mathcal{P}$ such that in its Ehrhart polynomial $i(\mathcal{P}, t)$ the coefficients of $t^{i_1}, \ldots, t^{i_k}$ are negative, while all remaining coefficients are positive? This problem was proposed by Hibi, Higashitani, Tsuchiya, and Yoshida. In this paper, we first construct a class of simplices $\mathcal{S}_d(m)$ whose Ehrhart polynomial has leading coefficient $m$ and all other coefficients fixed positive constants. Then, using the Cartesian product of $\mathcal{S}_d(m)$ and the Reeve tetrahedron, we obtain the first complete solution to the sign pattern problem. Finally, while attacking the sign pattern problem, we discovered a fast algorithm for computing the $h^*$-polynomial of a class of simplices $\Delta(0,q)$. This algorithm is crucial for constructing the simplices $\mathcal{S}_d(m)$.

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.

Desk Editor's Note private letter to a colleague

The product with the fixed Reeve tetrahedron only produces one sign pattern, so the main claim fails.

read the letter

The construction does not deliver a complete solution to the sign pattern problem. The Cartesian product with the fixed Reeve tetrahedron only realizes the sign pattern coming from the Reeve polynomial itself, shifted by the degree of S_d(m). The new part is the simplices S_d(m). These have Ehrhart polynomials where the leading coefficient is the parameter m and all lower coefficients are positive constants independent of m. That is a clean explicit family and seems to be new. The fast algorithm for h*-polynomials of the class Delta(0,q) is also a practical addition that could see use. The problem is in the second step. When you take the product P = S_d(m) × R, where R is the Reeve tetrahedron, the Ehrhart polynomial multiplies: i(P,t) = i(S_d(m),t) * i(R,t). Since i(S_d(m),t) = m t^d + c_{d-1} t^{d-1} + ... + c_0 with all c_i >0 fixed, the product is m t^d * r(t) + (lower poly) * r(t). For large m the dominant term is m times t^d r(t), so the signs of the coefficients in i(P,t) are eventually the same as the signs in t^d r(t). The Reeve tetrahedron has a fixed Ehrhart polynomial with its own sign pattern, so only that one pattern (or variations if some coeffs zero) can appear. But the sign pattern problem asks for every possible subset of coefficients to be negative. One fixed pattern is not enough. The abstract invokes that the product preserves independent control when m is large, but the algebra shows it does not. This is a load-bearing issue for the main theorem. The paper is aimed at people working on Ehrhart polynomials and combinatorial geometry. Readers looking for a solution to the open problem will be disappointed. The S_d(m) family might still be worth citing if the formulas check out, but the overall result does not hold. I would not send this to referees; the flaw is clear from the construction method described.

Referee Report

1 major / 1 minor

Summary. The paper constructs a family of d-dimensional simplices S_d(m) (d ≥ 3) whose Ehrhart polynomials have leading coefficient m and all lower-degree coefficients equal to fixed positive constants independent of m. It then asserts that the Cartesian product of any such simplex with the Reeve tetrahedron yields a polytope whose Ehrhart polynomial realizes an arbitrary prescribed sign pattern (any choice of negative coefficients in positions 1 through d−2, with all others positive). A secondary contribution is a fast algorithm for the h*-polynomials of the simplices Δ(0,q).

Significance. If the central construction were valid, the result would constitute the first explicit and complete solution to the sign-pattern problem posed by Hibi, Higashitani, Tsuchiya, and Yoshida, a notable advance in Ehrhart theory. The explicit parametric family S_d(m) and the algorithm for h*-polynomials of Δ(0,q) would also be useful technical tools. The manuscript does not, however, contain machine-checked proofs or reproducible code.

major comments (1)

[Abstract and construction section] Abstract and construction section: the claim that the Cartesian product S_d(m) × R (R the Reeve tetrahedron) realizes every possible sign pattern is incorrect. The Ehrhart polynomial of the product is i(S_d(m),t) · i(R,t) = (m t^d + p(t)) · r(t), where p(t) has fixed positive coefficients and r(t) is the fixed cubic Ehrhart polynomial of R. For large m the signs of the resulting coefficients are those induced by the shifted polynomial t^d r(t) (modulo the fixed lower-order contribution p(t)r(t)). Because r(t) is fixed, only one specific sign pattern (or a small finite family if some coefficients of r vanish) can appear; this cannot produce every subset of negative positions in {1,…,d−2}.

minor comments (1)

[Abstract] Abstract: 'well-konwn' is a typographical error for 'well-known'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a substantive issue in our central claim. We respond to the major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Abstract and construction section] Abstract and construction section: the claim that the Cartesian product S_d(m) × R (R the Reeve tetrahedron) realizes every possible sign pattern is incorrect. The Ehrhart polynomial of the product is i(S_d(m),t) · i(R,t) = (m t^d + p(t)) · r(t), where p(t) has fixed positive coefficients and r(t) is the fixed cubic Ehrhart polynomial of R. For large m the signs of the resulting coefficients are those induced by the shifted polynomial t^d r(t) (modulo the fixed lower-order contribution p(t)r(t)). Because r(t) is fixed, only one specific sign pattern (or a small finite family if some coefficients of r vanish) can appear; this cannot produce every subset of negative positions in {1,…,d−2}.

Authors: We agree with the referee's analysis. The coefficients of the product Ehrhart polynomial are an affine function of the parameter m. Consequently, as m ranges over the positive integers, only finitely many distinct sign patterns can arise. This construction therefore realizes only a limited collection of sign patterns rather than every possible choice of negative coefficients in positions 1 through d-2. We will revise the abstract, introduction, and construction section to remove the claim of a complete solution and to state precisely which sign patterns are achieved by the family S_d(m) × R. We are investigating whether a modified construction (for example, products involving several Reeve tetrahedra with different parameters or additional simplices) can be used to obtain the full result; any such extension will be presented in a future version or a follow-up note. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction is explicit and independent of target sign patterns

full rationale

The paper defines an explicit family of simplices S_d(m) whose Ehrhart polynomial is stated to have leading coefficient m with all lower coefficients fixed positive constants independent of m. It then forms the Cartesian product with the fixed Reeve tetrahedron and claims the resulting Ehrhart polynomial realizes arbitrary sign patterns for sufficiently large m. No step reduces a claimed result to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is itself unverified. The derivation chain relies on standard multiplicative properties of Ehrhart polynomials under products and an explicit parametrization of S_d(m), both of which are presented as constructed rather than assumed or fitted to the target sign patterns. This is the normal case of a self-contained explicit construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The construction rests on standard facts about Ehrhart polynomials of products and the known Ehrhart polynomial of the Reeve tetrahedron; no new free parameters beyond the integer m (chosen large) or invented entities are introduced.

free parameters (1)

m
Positive integer parameter controlling the leading coefficient of the Ehrhart polynomial of S_d(m); chosen sufficiently large to realize desired signs.

axioms (2)

standard math Ehrhart polynomials multiply under Cartesian product of polytopes.
Invoked when forming the product with the Reeve tetrahedron.
domain assumption The Reeve tetrahedron has an Ehrhart polynomial with known negative coefficients.
Used as the source of negative signs in the construction.

pith-pipeline@v0.9.0 · 5736 in / 1357 out tokens · 47615 ms · 2026-05-25T04:27:46.369100+00:00 · methodology

Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)