Multivariate discriminant and iterated resultant

Abstract

In this paper, we study the relationship between iterated resultant and multivariate discriminant. We show that, for generic form $f(X_n)$ with even degree $d$, if the polynomial is squarefreed after each iteration, the multivariate discriminant $\hat{I}”(f)$ is a factor of the squarefreed iterated resultant. In fact, we find a factor $Hp(f,[x_1,\dots ,x_n])$ of the squarefreed iterated resultant, and prove that the multivariate discriminant $\hat{I}”(f)$ is a factor of $Hp(f,[x_1,\dots ,x_n])$. Moreover, we conjecture that $Hp(f,[x_1,\dots ,x_n])=\hat{I}”(f)$ holds for generic form $f$, and show that it is true for generic trivariate form $f(x,y,z)$.