Macaulay-like marked bases

We define marked sets and bases over a quasi-stable ideal $\mathfrak j$ in a polynomial ring on a Noetherian $K$-algebra, with $K$ a field of any characteristic. The involved polynomials may be non-homogeneous, but their degree is bounded from above by the maximum among the degrees of the terms in the Pommaret basis of $\mathfrak j$ and a given integer $m$. Due to the combinatorial properties of quasi-stable ideals, these bases behave well with respect to homogenization, similarly to Macaulay bases. We prove that the family of marked bases over a given quasi-stable ideal has an affine scheme structure, is flat and, for large enough $m$, is an open subset of a Hilbert scheme. Our main results lead to algorithms that explicitly construct such a family. We compare our method with similar ones and give some complexity results.