Decomposable forms and automorphisms.
Let \(\mathcal{F}_{d}\) be the set of decomposable forms of degree \(d>0\) over an algebraically closed field \(K\) with variables in a \(K\)-vector space \(V\). Then \(\text{GL}(V) \) acts on \(\mathcal{F}_{d}\) via \(\gamma \cdot f=f\circ \gamma ^{-1}\), \(\gamma \in \text{GL}(V)\), \(f\in \mathcal{F}_{d}.\) For a given \(f\), \(\Aut(f) \) is defined to be the stabilizer of this action. This paper studies the group \( \Aut(f) \) for certain classes of forms. As \(f\) is decomposable we may write \(f=l_{1}l_{2}\cdots l_{d}\) for \(l_{i}\in V^{\ast }.\) Writing \(l=( l_{1},\ldots ,l_{d}) \) there is a map \( \widetilde{\rho}_{l}:\Aut(f) \to \Sigma _{d},\) where \(\Sigma _{d}\) is the symmetric group on \(d\) letters. While this map depends on the choice of decomposition \(l\), the author proves that, for \( m=( m_{1},\ldots ,m_{d}) \) another decomposition of \(f\), the maps \(\widetilde{\rho}_{l}\) and \(\widetilde{\rho}_{m}\) are conjugate in \(\Sigma _{d}.\) Let \(\Gamma _{l}( f) \) be the image of \(\Aut(f) \) under \(\widetilde{\rho}_{l}\). Then \(\Aut(f) \) is the semidirect product of \(\mu _{d}\) (the \(d\)th roots of unity) with \(\Gamma(f) \) where \(\Gamma(f) =\Gamma _{l}( f) \) for any decomposition \(l\). Thus the computation of \(\Aut(f) \) is equivalent to the computation of \(\Gamma(f) .\) In the second section, the author does some computations of \(\Gamma(f) \) in special cases. For a non-degenerate form \(f\) of degree \(d>0\), let \(k\) denote the number of variables. If \(k=2\) and \(d\geq 4\) then any nontrivial \(\tau \in \Gamma(f) \) is the product of \(( d-i) /t\) cycles of length \( t\) and \(i\) fixed points for some \(t\) and for \(0\leq i\leq 2.\) If \(k=3\) and \( d\geq 5\) then any nontrivial \(\tau \in \Gamma(f) \) is either the product of \(( d-i) /t\) \(t\)-cycles for \(0\leq i\leq 3\), or \((d-2-i)/t\) \(t\)-cycles, one \(2\)-cycle, and \(i\) fixed points for \( i=0\) or \(1.\) An element of \(\Aut(f) \) not in \(\mu _{d}\) is called nontrivial. Thus a nondegenerate form has a nontrivial automorphism if and only if \(\Gamma(f) \) is nontrivial. If all automorphisms of \(f\) are trivial we say \(f\) is generic. The remainder of the paper is devoted to determining when \(f\) is generic. It is shown that, in the case where \(k=2\) and \(d=4\) we have that every nondegenerate form has \(| \Gamma(f) | \) equal to 4, 8, or 12, and hence \(f\) is not generic. Cases are given to show that each of the three possible orders are in fact achieved for various \(f\). In the final section, the author proves, for \(k=2\) and \(d\geq 5\) and for \( k\geq 3\) and \(d\geq k+2,\) that ``almost all forms'' are generic. Specifically, it is shown that the set of \(( l_{1},\ldots ,l_{d}) \in(V^{\ast }) ^{d}\) for which \(f=l_{1}\cdots l_{d}\) is dense (in the Zariski topology) inside \(( V^{\ast }) ^{d}.\)
scientific article; zbMATH DE number 1915504