Galois groups of some vectorial polynomials (Q2719055)

Let \(q>1\) be a power of a prime \(p\), let \(k_q\) be an extension of the \(q\)-element field GF\((q)\). In the previous papers of the first author [Trans. Am. Soc. 348, 1555-1577 (1996; Zbl 0860.12001), Proc. Am. Math. Soc. 124, 2967-2976 (1996; Zbl 0866.12004), and 124, 2977-2991 (1996; Zbl 0866.12005)] the Galois groups of some vectorial \(q\)-polynomials over the fields \(k_q(X), k_q(X,T)\) were described. Under certain assumptions on the field \(k_q\) and the numbers \(q\) and \(m\), the odd dimensional unitary, symplectic, and even dimensional negative orthogonal groups SU\((2m-1,\sqrt{q})\), Sp\((2m,q)\) and \(\Omega^-(2m,q)\), respectively, appeared as Galois groups of the examined polynomials. NEWLINENEWLINENEWLINEIn the papers of the first author and \textit{P. A. Loomis} [Proc. Am. Math. Soc. 126, 1885-1896 (1998; Zbl 0889.12005), Contemp. Math. 256, 63-76 (1999; Zbl 0982.12002)] it was shown that one could obtain the same groups under weaker assumptions on \(k_q, q\) and \(m.\) NEWLINENEWLINENEWLINEThe main aim of the paper under review is to find vectorial \(q\)-polynomials, the Galois groups of which are the remaining classical groups. The authors show that the Galois groups of the following vectorial \(q\)-polynomials NEWLINE\[NEWLINEE^\ddag(Y)=Y^{q^{2m}}-X^{q'-1}Y^{q^{2m-1}}+X^{q'}Y^{q^m}-Y^q+X^{q'-1}Y,NEWLINE\]NEWLINE NEWLINE\[NEWLINEE^\circ(Y)=Y^{q^{2m+1}}-X^{q-1}Y^{q^{2m-1}}+X^qY^{q^{m+1}}-X^qY^{q^{m}}+Y^{q^2}-X^{q-1}Y,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{multlined} E^+(Y)=Y^{q^{2m}}-Y^{q^{2m-1}}-X^{q-1}Y^{q^{2m-2}} +X^{q-1}Y^{q^{2m-3}}+X^qY^{q^{m}}\\ +Y^{q^{3}}-Y^{q^{2}}-X^{q-1}Y^q+X^{q-1}Y\end{multlined}NEWLINE\]NEWLINE over \(k_q(X)\) are even dimensional unitary, odd dimensional orthogonal, and even dimensional positive orthogonal groups U\((2m,q')\), SO\((2m+1,q)\), and SO\(^+(2m,q)\), respectively, where \(q=(q')^2\) in the unitary case.