Componentwise linear ideals in Veronese rings

In this article, we study the componentwise linear ideals in the Veronese subrings of $R=K[x_1,\ldots,x_n]$. If char$(K)=0$, then we give a characterization for graded ideals in the $c^{th}$ Veronese ring $R^{(c)}$ to be componentwise linear. This characterization is an analogue of that over $R$ due to Aramova, Herzog and Hibi in \cite{ah00} and Conca, Herzog and Hibi in \cite{chh04}.