Symmetries of polynomials
Let \(Q(x,y) = \sum_{i=0}^n a_ix^i y^{n-i}\) be a binary form of degree~\(n\) with real or complex coefficients \(a_i\). Classical invariant theory deals with the invariants of binary forms under the action of SL\(_2\). The article under review takes a different point of view. For \(Q(x,y)\) as above with fixed coefficients \(a_i\), the goal is to determine the symmetry group of \(Q(x,y)\), i.e., the group of all invertible \(2 \times 2\) matrices with entries \(\alpha,\beta,\gamma,\delta\) such that \(Q(\alpha x + \beta y,\gamma x + \delta y) = Q(x,y)\). The authors present a new algorithm for finding the symmetry group. An implementation of the algorithm in Maple is described. The algorithm relies on certain covariants, such as the Hessian \(H = Q_{x x} Q_{y y} - Q_{x y}^2\) and the Jacobian \(J[Q,H] = Q_x H_y - Q_y H_x\). From these (and one other) covariants, a set of equations is derived whose solutions correspond to the symmetries of \(Q(x,y)\). It is well-known that the projective symmetry group is either a two-parameter group, a one-parameter group, or a finite group from a list given by \textit{H. F. Blichfeldt} (1917). The Maple code decides which of these cases applies for a given binary form.
scientific article; zbMATH DE number 1471043