Combinatorial meaning of the coefficients of a Hilbert polynomial

Abstract

In [1] Abhyankar defines an idealI(p, a) generated by certain minors of a matrixX, the entries ofX being independent indeterminates, and proves that the Hubert function ofI(p, a) coincides with its Hilbert polynomialF(V) and obtains it in the form

$$F(V) = \sum\limits_{D \geqslant 0} {( - 1)} ^D F_D (m,p,a)\left( \begin{gathered} c - D + V \hfill \\ V \hfill \\ \end{gathered} \right)$$

. He also proves thatF(V) is the number of certain “indexed” monomials of degreeV in the entries ofX and that the coefficientsF _D (m,p,a) are non-negative integers and asks for their combinatorial meaning. In this paper we characterize the indexed monomials in terms of certain sets of lattice paths, called frames, and prove that the coefficientsF _D (m, p, a) count certain families of such frames.