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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

    Examples illustrating some aspects of the weak Deligne-Simpson pro blem

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    We consider the variety of (p+1)(p+1)-tuples of matrices AjA_j (resp. MjM_j) from given conjugacy classes cjβŠ‚gl(n,C)c_j\subset gl(n,{\bf C}) (resp. CjβŠ‚GL(n,C)C_j\subset GL(n,{\bf C})) such that A1+...+Ap+1=0A_1+... +A_{p+1}=0 (resp. M1...Mp+1=IM_1... M_{p+1}=I). This variety is connected with the weak {\em Deligne-Simpson problem: give necessary and sufficient conditions on the choice of the conjugacy classes cjβŠ‚gl(n,C)c_j\subset gl(n,{\bf C}) (resp. CjβŠ‚GL(n,C)C_j\subset GL(n,{\bf C})) so that there exist (p+1)(p+1)-tuples with trivial centralizers of matrices Aj∈cjA_j\in c_j (resp. Mj∈CjM_j\in C_j) whose sum equals 0 (resp. whose product equals II).} The matrices AjA_j (resp. MjM_j) are interpreted as matrices-residua of Fuchsian linear systems (resp. as monodromy operators of regular linear systems) on Riemann's sphere. We consider examples of such varieties of dimension higher than the expected one due to the presence of (p+1)(p+1)-tuples with non-trivial centralizers; in one of the examples the difference between the two dimensions is O(n).Comment: Research partially supported by INTAS grant 97-164
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