I have been solving systems of polynomial equations by forming the Macaulay matrix of different degrees and computing its null space. If the degree is large enough, namely at or above the degree of regularity, the null space is spanned by Vandermonde vectors in the roots and Macaulay matrices of higher degrees will have a null space of the same dimension. For the solution, I thus need to know this degree of regularity. This is a very hard problem, but in practice (for generic systems), I notice that it always equals $\sum_{s=0}^S (d_s-1)$ with $S$ the number of equations and $d_s$ the degree of the $s$th polynomial in the system. The Macaulay bound is an upper bound on the degree of regularity and equals $1+\sum_{s=0}^S (d_s-1)$, so $1$ higher than what I encounter generically. I see why the degree of regularity could be lower, but not why it could be higher. Can someone give me an example system where the degree of regularity is exactly equal to the upper bound?
