Gr\"obner Bases of Generic Ideals

Let $I=(f_1,\dots ,f_n)$ be a homogeneous ideal in the polynomial ring $K[x_1,\dots ,x_n]$ over a field $K$ generated by generic polynomials. Using an incremental approach based on a method by Gao, Guan and Volny, and properties of the standard monomials of generic ideals, we show how a Gr"obner basis for the ideal $(f_1,\dots ,f_i)$ can be obtained from that of $(f_1, \dots, f_{i-1})$\mathrm{.}If$deg f_i = d_i$, we are able to give a complete description of the initial ideal of $I$ in the case where $d_i \geq \left(\sum_{j=1}^{i-1}d_j\right) - i -1$. It was conjectured by Moreno-Soc'ias that the initial ideal of $I$ is almost reverse lexicographic, which implies a conjecture by Fr"oberg on Hilbert series of generic algebras. As a result, we obtain a partial answer to Moreno-Soc'ias Conjecture: the initial ideal of $I$ is almost reverse lexicographic if the degrees of generators satisfy the condition above. This result improves a result by Cho and Park. We hope this approach can be strengthened to prove the conjecture in full.