A Property of Discriminants

Abstract

For the family P(x, a) := xⁿ + a₁x^n− 1 + ⋯ + a_n of complex polynomials in the variable x, we study its discriminantR := Res(P, P^′, x), $R\in \mathbb {C}[a]$, a = (a₁,…, a_n). When R is regarded as a polynomial in a_k, one can consider its discriminant $\tilde {D}_{k}:=\text {Res}(R,\partial R/\partial a_{k},a_{k})$. We show that $\tilde {D}_{k}=c_{k}(a_{n})^{d(n,k)}{M_{k}^{2}}{T_{k}^{3}}$, where $c_{k}\in \mathbb {Q}^{\ast }$, d(n, k) := min(1, n − k) + max(0, n − k − 2), the polynomials $M_{k},T_{k}\in \mathbb {C}[\hat {a}^{k}]$ have integer coefficients, $\hat {a}^{k}=(a_{1},{\ldots } ,a_{k-1},a_{k + 1},{\ldots } ,a_{n})$, the sets {M_k = 0} and {T_k = 0} are the projections in the space of the variables $\hat {a}^{k}$ 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 ≤ k ≤ n − 1 and P_n := P^′. One has $T_{k}=\text {Res}(P_{k},P_{k}^{\prime },x)$ for k ≠ n − 1 and $T_{n-1}=\text {Res}(P_{n-1},P_{n-1}^{\prime },x)/a_{n}$.