  For each $\nu\in \{1,2\}$, we examine the set $\mathscr{U}_{x,y}^\nu$ of all
formal bivariate Laurent series $f(x,y)=\sum_{i,j\in\mathbb{Z}}c_{ij}x^iy^j$
such that $c_{i,j}=c_{-i,j}$ and $c_{i,j}\ge c_{i+\nu, j}\ge 0$ for all
non-negative integers $i$ and all integers $j$. We show that the set
$\mathscr{U}_{x, y}^\nu$ forms a commutative semiring under the operations of
addition and multiplication of formal Laurent series. This result is used to
establish the unimodality of sequences involving Gauss polynomials and certain
refined color partitions. In particular, we solve an open problem posed by
Andrews on the unimodality of generalized Gauss polynomials and establish an
unimodal result for a statistic of plane partitions. We also establish many
unimodal results for rank statistics in partition theory, including the rank
statistics of concave and convex compositions studied by Andrews, as well as
certain unimodal sequences studied by Kim-Lim-Lovejoy. Additionally, we
establish the unimodality of the Betti numbers and Gromov-Witten invariants of
certain Hilbert schemes of points.
