Bézout's theorem and ideals of terminal forms (Q984626)

Let \(K\) be a field and \(f\in K[x_1, \ldots, x_n]\) a polynomial of degree \(d\). The terminal form \(tm(f)\) is the homogeneous part of degree \(d\) of \(f\). For an ideal \(I\) the terminal ideal \(tm(I)\) is the ideal generated by the terminal forms of the elements of \(I\). Let \(I=\langle f_1, \ldots, f_n\rangle\) be a zero--dimensional ideal and assume that \(\langle x_1, \ldots, x_n\rangle^N\subset \langle tm(f_1), \ldots, tm(f_n)\rangle\) for some \(N\). It is proved that \(tm(I)=\langle tm(f_1), \ldots, tm(f_n)\rangle\). This result is equivalent to Bézout's theorem.