Hilbert series and degrees of regularity of Oil & Vinegar and mixed quadratic systems

In this paper, we analyze the algebraic invariants for two classes of multivariate quadratic systems: systems made by oil and vinegar quadratic polynomials and systems made by both oil and vinegar polynomials and fully-quadratic ones. For such systems, we explicitly compute the Hilbert series in the homogeneous case, and we also give bounds on the degree of regularity, solving degree and first fall degree. Such degrees can be relevant to compute the complexity of solving those systems and to estimate their cryptographic security.