Nonsymmetric generic matrix equations

Let (Ai)0≤i≤k(Ai)0≤i≤k be generic matrices over QQ, the field of rational numbers. Let K=Q(E)K=Q(E), where E   denotes the entries of the (Ai)i(Ai)i, and let K¯ be the algebraic closure of K  . We show that the generic unilateral equation AkXk+⋯+A1X+A0=0nAkXk+⋯+A1X+A0=0n has (nkn) solutions X∈Mn(K¯). Solving the previous equation is equivalent to solving a polynomial of degree kn  , with Galois group SknSkn over K  . Let (Bi)i≤k(Bi)i≤k be fixed n×nn×n matrices with entries in a field L  . We show that, for a generic C∈Mn(L)C∈Mn(L), a polynomial equation g(B1,⋯,Bk,X)=Cg(B1,⋯,Bk,X)=C admits a finite fixed number of solutions and these solutions are simple. We study, when n=2n=2, the generic non-unilateral equations X2+BXC+D=02X2+BXC+D=02 and X2+BXB+C=02X2+BXB+C=02. We consider the unilateral equation Xk+Ck−1Xk−1+⋯+C1X+C0=0nXk+Ck−1Xk−1+⋯+C1X+C0=0n when the (Ci)i(Ci)i are n×nn×n generic commuting matrices; we show that every solution X∈Mn(K¯) commutes with the (Ci)i(Ci)i. When n=2n=2, we prove that the generic equation A1XA2X+XA3X+X2A4+A5X+A6=02A1XA2X+XA3X+X2A4+A5X+A6=02 admits 16 isolated solutions in M2(K¯), that is, according to Bézout's theorem, the maximum for a quadratic 2×22×2 matrix equation.