# Find the Picard Fuchs operator of a four parameter fundamental period

In my research, we have constructed a Calabi-Yau as a hypersurface of a toric variety and we could compute the fundamental period of the complex moduli, which is a series in four parameters. Say $$\varpi_0=1+\sum_{i,j,k,l}a_{ijkl} ~z_1^i z_2^j z_3^k z_4^l$$ We want to compute the Picard Fuchs operator, $\mathcal{L}$ which is a differential operator such that $$\mathcal{L}\varpi_0=0$$

But since there are four parameters, my computer could not find one, does anyone know any systematical way to construct such operators in four parameters? Any reference would be greatly appreciated.

• A standard way to handle this kind of problem is to compute the GKZ system first and then remove irrelevant part. See projecteuclid.org/euclid.cmp/1104288301 May 31 '16 at 4:09