For the family of polynomials in one variable P:=xn+a1βxnβ1+β―+anβ, nβ₯4, we consider its higher-order discriminant sets {D~mβ=0}, where D~mβ:=Res(P,P(m)), m=2, β¦,
nβ2, and their projections in the spaces of the variables ak:=(a1β,β¦,akβ1β,ak+1β,β¦,anβ). Set P(m):=βj=0nβmβcjβajβxnβmβj, Pm,kβ:=ckβPβxmP(m). We show that
Res(D~mβ,βD~mβ/βakβ,akβ)=Am,kβBm,kβCm,k2β, where Am,kβ=annβmβkβ,
Bm,kβ=Res(Pm,kβ,Pm,kβ²β) if 1β€kβ€nβm and
Am,kβ=anβmnβkβ, Bm,kβ=Res(P(m),P(m+1)) if nβm+1β€kβ€n. The equation Cm,kβ=0 defines the projection in the space of the
variables ak of the closure of the set of values of (a1β,β¦,anβ) for
which P and P(m) have two distinct roots in common. The polynomials
Bm,kβ,Cm,kββC[ak] are irreducible. The result is generalized
to the case when P(m) is replaced by a polynomial Pββ:=βj=0nβmβbjβajβxnβmβj, 0ξ =biβξ =bjβξ =0 for iξ =j