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Let  K  be a field of characteristic zero. The  K -vector space  V nof all binary formsof degree  n  with coefficients in  K  carries a natural action of the groupvia substitutions of variables  x  and  y . The invariant polynomialsunder the action of certain subgroups ofwere 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 inonly forIt appears to be a hopeless task to get much further along these lines. However, it is known that after inverting the  U -invariant  a0 , the algebra has a very special form, namely it is generated by algebraically independent polynomialstogether with In this note, we give an explicit set of such polynomialswhich are of degrees 2 and 3. We also extend these results to  U -invariants of several binary forms.

A new basis for the U -invariants of binary forms