Conjectural Hilbert series for generic quadratic ideals in four and five variables

Let $R_{n,r,s}=S_n/I^s$, where $I$ is generated by $r$ generic quadratic forms in $S_n=k[x_1,\ldots,x_n]$. The Hilbert series is written as $R_{n,r,s}(t)=\sum_{i\ge0}\dim_k(R_{n,r,s})_i t^i$, and the Hilbert coefficients are the coefficients in the eventual binomial-polynomial expression for $l(R_{n,r,s})$.

Generic quadratic Hilbert-series conjecture. For the indicated values of $n,r$, the following formulas hold:

• For $R_{4,5,s}$ and $s\ge4$,

$$R_{4,5,s}(t)=\sum_{i=0}^{2s-1}{i+3\choose3}t^i+10s^2-25s+35,$$

and $(e_0,e_1,e_2,e_3,e_4)=(16,12,21,35,35)$.

• For $R_{4,6,s}$ and $s\ge4$,

$$R_{4,6,s}(t)=\sum_{i=0}^{2s-1}{i+3\choose3}t^i+10s,$$

and $(e_0,e_1,e_2,e_3)=(16,12,1,10)$.

• For $R_{4,7,s}$ and $s\ge3$,

$$R_{4,7,s}(t)=\sum_{i=0}^{2s-1}{i+3\choose3}t^i,$$

and $(e_0,e_1,e_2)=(16,12,1)$.

• For $R_{5,9,s}$ and $s\ge4$,

$$R_{5,9,s}(t)=\sum_{i=0}^{2s-1}{i+4\choose4}t^i,$$

and $(e_0,e_1,e_2)=(32,32,6)$.

These formulas extend the few cases whose Hilbert series are known and concern the difficult problem of determining powers of ideals generated by generic quadrics. The source presents them as conjectural and gives no proof or resolution.