Multivariate discriminant and iterated resultant

TL;DR

This paper explores the mathematical relationship between iterated resultants and multivariate discriminants, establishing factorization properties and conjecturing their equality for generic forms, with proofs for specific cases.

Contribution

It demonstrates that for generic even-degree forms, the multivariate discriminant divides the iterated resultant and conjectures their equality in general, supported by proofs for trivariate forms.

Findings

Multivariate discriminant is a factor of the squarefree iterated resultant.

Proved the conjecture for generic trivariate forms.

Established a factorization relationship for generic forms.

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 $\Delta(f)$ is a factor of the squarefreed iterated resultant. In fact, we find a factor $Hp(f,[x_1,\ldots,x_n])$ of the squarefreed iterated resultant, and prove that the multivariate discriminant $\Delta(f)$ is a factor of $Hp(f,[x_1,\ldots,x_n])$. Moreover, we conjecture that $Hp(f,[x_1,\ldots,x_n])=\Delta(f)$ holds for generic form $f$, and show that it is true for generic trivariate form $f(x,y,z)$.