On higher-order discriminants

For the family of polynomials in one variable P := x(n) + a(1)x(n-1) + . . . + a(n), n >= 4, we consider its higher-order discriminant sets {(D) over tilde (m) = 0}, where (D) over tilde (m) := Res(P, P-(m)), m = 2, ..., n - 2, and their projections in the spaces of the variables a(k) := (a(1), ..., a(k-1) a(k+1), ..., a(n)). Set P-(m) := Sigma(n-m)(j=0) c(j)a(j) x(n-m-j), P-m(,)k( ):= c(k) P - x(m) P-(m). We show that Res((D) over tilde (m), partial derivative(D) over tilde (m))partial derivative a(k), a(k)) = A(m,k)B(m,k)C(m,k)(2), where A(m,k )= a(n)(n-m-k), B-m,B-k = Res(P-m,P-k, P-m,P-k' ) if 1 <= k <= n m and = A(m,k) = a(n-m)(n-k), B-m,B-k = Res(P-(m),P(m+1)) if n - m + 1 <= k <= n. The equation C-m,C-k = 0 defines the projection in the space of the variables a(k) of the closure of the set of values of (a(1), ..., a(n)) an) for which P and P((m) )have two distinct roots in common. The polynomials B-m,B-k, C-m,C-k is an element of C[a(k)] are irreducible. The result is generalized to the case when P((m) )is replaced by a polynomial P-* := Sigma(n-m)(j=0) b(j)a(j)x(n-m-j) 0 not equal b(i) not equal b(j) not equal 0 for i not equal j. (C) 2020 Elsevier Masson SAS. All rights reserved.