Ideals and quotients of diagonally quasi-symmetric functions

In 2004, J-C. Aval, F. Bergeron and N. Bergeron studied the algebra of diagonally quasi-symmetric functions DQSym in the ring Q[x,y] with two sets of variables. They made conjectures on the structure of the quotient Q[x,y]/< DQSym(+)>, which is a quasi-symmetric analogue of the diagonal harmonic polynomials. In this paper, we construct a Hilbert basis for this quotient when there are infinitely many variables i.e. x = x(1),x(2), and y = y(1),y(2), Then we apply this construction to the case where there are finitely many variables, and compute the second column of its Hilbert matrix.