Scott-type inequalities for trinomial $h^*$-polynomials

Abstract

Motivated by Scott's inequality for lattice polygons, we study normalized volume bounds for lattice polytopes $\mathcal{P}$ with $h^*$-polynomials $1+at^k+bt^\ell$, where $a,b\ge1$ and $1\le k<\ell$. For $\ell>2k$, the coefficients satisfy $a+1\le b$, and consequently ${\rm vol}(\mathcal{P})\le2b$. In the complementary range $k<\ell\le2k$, we establish the dimension-independent bound \[ {\rm vol}(\mathcal{P})\le(2b+1)(k+\ell). \] On the boundary $\ell=2k$ with $b\ge2$, the sharper Scott-type inequality ${\rm vol}(\mathcal{P})\le4k(b+1)$ holds, confirming a conjecture of Higashitani, Nill, and the author. Equality holds if and only if $k=1$ and $a=3b+3$. Moreover, when $\ell=2k$ and $k,b\ge2$, we obtain an upper bound depending only on $b$.