On lower bounds for the F-pure threshold of equigenerated ideals

Let $k$ be a field of positive characteristic and $R=k[x_0,\dots ,x_n]$. We consider ideals $I\subseteq R$ generated by homogeneous polynomials of degree $d$. Takagi and Watanabe proved that $\mathrm{f}\mathrm{p}\mathrm{t}(I)\ge \mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}(I)\mathrm{/}d$; we classify ideals $I$ for which equality is attained. Inspired by a result of de Fernex, Ein, and Mustaţă, we give a lower bound on $\mathrm{f}\mathrm{p}\mathrm{t}(I)$ in terms of the height of $\tau (I^{\mathrm{f}\mathrm{p}\mathrm{t}(I)})$.