In commutative algebra, people care about the Hilbert function of the associated graded ring is non-decreasing. As far as I know, motivation came from Sally's question in 1978 which is one dimensional Cohen-Macaulay ring with a small embedding dimension that has a non-decreasing Hilbert function or not.
But, I still don't get what is the advantage of having a non-decreasing Hilbert function, why people so care about it?
And what is the geometric difference between non-decreasing and decreasing functions?
