An application of the resultant to oscillatory integrals with polynomial phase (Q1378344)

In this brief and elegant note, the author obtains estimates for integrals of the form \[ \int_I \exp(ip(t)) dt, \] which depend on the polynomial \(p\) but not on the interval \(I\). If the derivatives \(p^{(i)}\) and \(p^{(j)}\) have no common (complex) zeros, then the integral may be bounded by a fixed multiple (depending on \(i\) and \(j\) and the degree of \(p\)) of \(| P| ^{-1/(n^2-ij-n)}\), where \(P\) is the resultant of \(p^{(i)}\) and \(p^{(j)}\), divided by \(a_n\), where \(p(t)=\sum_{j=0}^n a_kt^k\).