On the Behaviour of Stanley Depth under Variable Adjunction

Let $S=K[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ be a polynomial ring in $n$ variables over the field $K$. For integers $1\le t<n$ consider the ideal $I=(x_1,…,x_t)\cap(x_{t+1}, …,x_n)$in$S$. In this paper we bound from above the Stanley depth of the ideal $I^{\mathrm{\prime }}=(I,x_{n+1},\mathrm{.}\mathrm{.}\mathrm{.},x_{n+p})\subset S^{\mathrm{\prime }}=S[x_{n+1},\mathrm{.}\mathrm{.}\mathrm{.},x_{n+p}]$. We give similar upper bounds for the Stanley depth of the ideal $(I_{n,2},x_{n+1},\mathrm{.}\mathrm{.}\mathrm{.},x_{n+p})$, where $I_{n,2}$ is the square free Veronese ideal of degree 2 in $n$ variables.