Invariants of binary forms

Abstract

We work over the field of complex numbers. Consider the set V_n of binary forms of degree n, i.e., homogeneous polynomials of degree n in two variables. An invariant of V_n is a polynomial in the coefficients of a form in V_n whose value does not change under the action of the group of complex 2x2 matrices with determinant 1 on V_n. In 1893, Hilbert proved that the algebra of invariants of V_n is finitely generated for all n. However, finding explicit sets of generators is in general a difficult problem. They were known for binary forms of degree 8 and less than 7 since the 19th century, but for higher degrees the problem remained open. Over the last 25 years some progress was booked: generators for the invariants of binary forms of degree 7 were found by Dixmier and Lazard, and for the invariants of forms of degrees 9 and 10 by Brouwer and myself. This thesis is organised as follows. In Chapter 2 we introduce definitions and notation that will be used throughout this thesis. In Chapter 3 we present the computational methods that we use for finding generating invariants of binary forms. In Chapter 4 we find the generating invariants of binary forms of degree 2,3,...,10, and give explicit systems of parameters in all these cases. In Chapter 5 we review classical results regarding the invariants of several binary forms. We correct a result of Winter on the generating covariants of V_2+V_5 and results of Gundelfinger and Sylvester on the generating covariants of V_3+V_4. In Chapter 6, extending a result of Popov, we classify the modules whose algebras of invariants have homological dimension less than or equal to 15.