In this talk, I will describe a strong Briançon-Skoda type result (the integral closure of \(J^{n+k-1}\) is contained in \(J^k\), or a slight enlargement of \(J^k\)) which utilizes the Eagon-Northcott
or Buchsbaum-Eisenbud complex.

This result gives a new proof of Lipman-Sathaye's Briançon-Skoda result for regular rings, it gives the
precise version of Briançon-Skoda for generalizations of pseudo-rational rings (improving Lipman-Teissier's result and implying Aberbach-Huneke in equal characteristic), it also implies the tight closure, plus closure, and epf closure versions of the Briançon-Skoda theorem. It also implies effective Briancon-Skoda results for characteristic free versions of Du Bois singularities, generalizing work of Huneke-Watanabe and Wheeler-Zhang in the F-pure case.  Finally, we show how this result, plus Gabber's weak local unformization, can be used as the missing piece to solve Huneke's conjecture of uniform Briançon-Skoda
and uniform Artin-Rees for (reduced) quasi-excellent rings of finite Krull dimension.