A property of discriminants

TL;DR

This paper investigates the structure of discriminants of polynomial families, revealing a factorization involving polynomials that describe the loci of multiple roots, and provides explicit formulas for these factors.

Contribution

It introduces a detailed factorization of the discriminant's discriminant, identifying polynomials that characterize multiple root strata in polynomial families.

Findings

The discriminant of the discriminant factors as a product involving polynomials M_k and T_k.

Polynomials M_k and T_k define the loci of double and triple roots respectively.

Explicit formulas relate T_k to resultants of certain derived polynomials P_k.

Abstract

For the family $P:=x^n+a_1x^{n-1}+\cdots +a_n$ of complex polynomials in the variable $x$ we study its {\em discriminant} $R:=$Res$(P,P',x)$, $R\in \mathbb{C}[a]$, $a=(a_1,\ldots ,a_n)$. When $R$ is regarded as a polynomial in $a_k$, one can consider its discriminant $\tilde{D}_k:=$Res$(R,\partial R/\partial a_k,a_k)$. We show that $\tilde{D}_k=c_k(a_n)^{d(n,k)}M_k^2T_k^3$, where $c_k\in \mathbb{Q}^*$, $d(n,k):=\min (1,n-k)+\max (0,n-k-2)$, the polynomials $M_k,T_k\in \mathbb{C}[a^k]$ have integer coefficients, $a^k=(a_1,\ldots ,a_{k-1},a_{k+1},\ldots ,a_n)$, the sets $\{ M_k=0\}$ and $\{ T_k=0\}$ are the projections in the space of the variables $a^k$ of the closures of the strata of the variety $\{ R=0\}$ on which $P$ has respectively two double roots or a triple root. Set $P_k:=P-xP'/(n-k)$ for $1\leq k\leq n-1$ and $P_n:=P'$. One has $T_k=${\rm Res}$(P_k,P_k',x)$ for $k\neq n-1$ and $T_{n-1}=${\rm Res}$(P_{n-1},P_{n-1}',x)/a_n$.