Scott-type inequalities for trinomial h^*-polynomials

Abstract

Motivated by Scott's inequality for lattice polygons, we study normalized volume bounds for lattice polytopes 𝒫 with h^*-polynomials 1+at^k+bt^ℓ, where a,b≥1 and 1≤ k<ℓ. For ℓ>2k, the coefficients satisfy a+1≤ b, and consequently vol(𝒫)≤2b. In the complementary range k<ℓ≤2k, we establish the dimension-independent bound vol(𝒫)≤(2b+1)(k+ℓ). On the boundary ℓ=2k with b≥2, the sharper Scott-type inequality vol(𝒫)≤4k(b+1) holds, confirming a conjecture of Higashitani, Nill, and the author. Equality holds if and only if k=1 and a=3b+3. Moreover, when ℓ=2k and k,b≥2, we obtain an upper bound depending only on b.

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BibTeX RIS

BibTeX

@misc{tsuchiya2026scotttype,
  title = {{Scott-type inequalities for trinomial $h^*$-polynomials}},
  author = {Akiyoshi Tsuchiya},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35287},
  eprint = {2610.35287},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35287},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Scott-type inequalities for trinomial h^*-polynomials
AU  - Akiyoshi Tsuchiya
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35287
UR  - https://arxiv.org/abs/2610.35287
AB  - Motivated by Scott's inequality for lattice polygons, we study normalized volume bounds for lattice polytopes 𝒫 with h^*-polynomials 1+at^k+bt^ℓ, where a,b≥1 and 1≤ k<ℓ. For ℓ>2k, the coefficients satisfy a+1≤ b, and consequently vol(𝒫)≤2b. In the complementary range k<ℓ≤2k, we establish the dimension-independent bound vol(𝒫)≤(2b+1)(k+ℓ). On the boundary ℓ=2k with b≥2, the sharper Scott-type inequality vol(𝒫)≤4k(b+1) holds, confirming a conjecture of Higashitani, Nill, and the author. Equality holds if and only if k=1 and a=3b+3. Moreover, when ℓ=2k and k,b≥2, we obtain an upper bound depending only on b.
ER  -

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