On the regularity of \(p\)-Borel ideals (Q2716098)

Let \(K\) be a field of characteristic \(p>0\). Let \(k\) and \(l\) be non-negative integers with \(p\)-adic expansion \(k=\sum k_ip^i\) and \(l=\sum l_i p^i\).NEWLINENEWLINENEWLINE\(k\leq_pl\) if \(k_i\leq l_i\) for all \(i\).NEWLINENEWLINENEWLINEA monomial ideal \(I\subset K[x_1, \dots, x_n]\) is called \(p\)-Borel if for each monomial \(u\in I\), \(u=\prod x_i^{\mu_i}\) one has \((x_i/x_j)^\nu u\in I\) for all \(i,j\) with \(1\leq i<j\leq u\) and all \(\nu\leq_p \mu_j\). A conjecture of \textit{K. Pardue} on the regularity of principal \(p\)-Borel ideals is proved.