The Charney-Davis conjecture for simple thin polyominoes

TL;DR

This paper proves that Gorenstein rings derived from simple thin polyominoes satisfy the Charney-Davis conjecture by analyzing their Hilbert series and establishing a specific inequality.

Contribution

It establishes the Charney-Davis conjecture for Gorenstein rings associated with simple thin polyominoes, linking combinatorial properties to algebraic invariants.

Findings

The inequality $(-1)^{loor{rac{ ext{deg} h_R(t)}{2}}}h_R(-1) \\geq 0$ holds for Gorenstein rings.

Gorenstein rings from simple thin polyominoes satisfy the Charney-Davis conjecture.

The Hilbert series analysis confirms the conjecture in this specific combinatorial setting.

Abstract

Let $\mathcal{P}$ be a simple thin polyomino and $\Bbbk$ a field. Let $R$ be the toric $\Bbbk$-algebra associated to $\mathcal{P}$. Write the Hilbert series of $R$ as $h_{R}(t)/(1-t)^{\dim(R)}$. We show that $$(-1)^{\left\lfloor{\frac{\mathrm{deg} h_R(t)}{2}}\right\rfloor}h_{R}(-1) \geq 0$$ if $R$ is Gorenstein. This shows that the Gorenstein rings associated to simple thin polyominoes satisfy the Charney-Davis conjecture.