A new basis for the U-invariants of binary forms

Abstract

Listen

Translate article

Let K be a field of characteristic zero. The K-vector space Vn of all binary forms of degree n with coefficients in K carries a natural action of the group via substitutions of variables x and y. The invariant polynomials under the action of certain subgroups of were studied intensively in the 19th century. If U is the subgroup of the upper triangular unipotent matrices, with the aid of heavy computer calculations, we know the algebra of U-invariant polynomials in only for It appears to be a hopeless task to get much further along these lines. However, it is known that after inverting the U-invariant a 0, the algebra has a very special form, namely it is generated by algebraically independent polynomials together with In this note, we give an explicit set of such polynomials which are of degrees 2 and 3. We also extend these results to U-invariants of several binary forms.