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    On higher-order discriminants

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    For the family of polynomials in one variable P:=xn+a1xnβˆ’1+β‹―+anP:=x^n+a_1x^{n-1}+\cdots +a_n, nβ‰₯4n\geq 4, we consider its higher-order discriminant sets {D~m=0}\{ \tilde{D}_m=0\}, where D~m:=\tilde{D}_m:=Res(P,P(m))(P,P^{(m)}), m=2m=2, …\ldots, nβˆ’2n-2, and their projections in the spaces of the variables ak:=(a1,…,akβˆ’1,ak+1,…,an)a^k:=(a_1,\ldots ,a_{k-1},a_{k+1},\ldots ,a_n). Set P(m):=βˆ‘j=0nβˆ’mcjajxnβˆ’mβˆ’jP^{(m)}:=\sum _{j=0}^{n-m}c_ja_jx^{n-m-j}, Pm,k:=ckPβˆ’xmP(m)P_{m,k}:=c_kP-x^mP^{(m)}. We show that Res(D~m,βˆ‚D~m/βˆ‚ak,ak)=Am,kBm,kCm,k2(\tilde{D}_m,\partial \tilde{D}_m/\partial a_k,a_k)= A_{m,k}B_{m,k}C_{m,k}^2, where Am,k=annβˆ’mβˆ’kA_{m,k}=a_n^{n-m-k}, Bm,k=B_{m,k}=Res(Pm,k,Pm,kβ€²)(P_{m,k},P_{m,k}') if 1≀k≀nβˆ’m1\leq k\leq n-m and Am,k=anβˆ’mnβˆ’kA_{m,k}=a_{n-m}^{n-k}, Bm,k=B_{m,k}=Res(P(m),P(m+1))(P^{(m)},P^{(m+1)}) if nβˆ’m+1≀k≀nn-m+1\leq k\leq n. The equation Cm,k=0C_{m,k}=0 defines the projection in the space of the variables aka^k of the closure of the set of values of (a1,…,an)(a_1,\ldots ,a_n) for which PP and P(m)P^{(m)} have two distinct roots in common. The polynomials Bm,k,Cm,k∈C[ak]B_{m,k},C_{m,k}\in \mathbb{C}[a^k] are irreducible. The result is generalized to the case when P(m)P^{(m)} is replaced by a polynomial Pβˆ—:=βˆ‘j=0nβˆ’mbjajxnβˆ’mβˆ’jP_*:=\sum _{j=0}^{n-m}b_ja_jx^{n-m-j}, 0β‰ biβ‰ bjβ‰ 00\neq b_i\neq b_j\neq 0 for iβ‰ ji\neq j
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