Classical invariant theory, by Peter Olver, London Mathematical Society Student

Classical invariant theory was a hot topic in the 19 century and in the beginning of the 20 century. The book under review is an attempt to revive this beautiful subject. In our new era of computers and new interest in computational aspects, it is certainly worthwhile to recall the constructive methods of the 19 century invariant theorists. In classical invariant theory one studies polynomials and their intrinsic properties. The book mostly deals with polynomials in one variable, or rather, homogeneous polynomials in two variables which are called binary forms. A first example is treated in Chapter 1. Let Q(x) = ax +bx+c be the quadratic polynomial (throughout the book, coefficients are always either real or complex). As we learned in high school, an important characteristic of a quadratic polynomial is the discriminant ∆ = b − 4ac. The discriminant tells us for example whether there are 0, 1 or 2 real solutions, and it is invariant under translation x 7→ x + α. We can view Q as a binary form of degree 2 by writing Q(x, y) = yQ(xy ) = ax + bxy + cy. We now have more symmetries available, namely the projective transformations