On Hilbert polynomial of certain determinantal ideals

Let X = ( X i j ) be an m ( 1 ) by m ( 2 ) matrix whose entries X i j , 1 ≤ i ≤ m ( 1 ) , 1 ≤ j ≤ m ( 2 ) ; are indeterminates over a field K . Let K [ X ] be the polynomial ring in these m ( 1 ) m ( 2 ) variables over K . A part of the second fundamental theorem of Invariant Theory says that the ideal I [ p + 1 ] in K [ X ] , generated by ( p + 1 ) by ( p + 1 ) minors of X is prime. More generally in [1], Abhyankar defines an ideal I [ p + a ] in K [ X ] , generated by different size minors of X and not only proves its primeness but also calculates the Hilbert function as well as the Hilbert polynomial of this ideal. The said Hilbert polynomial is completely determined by certain integer valued functions F D ( m , p , a ) . In this paper we prove some important properties of these integer valued functions.