Suppose we have the data points $(w,x,y,z) \bmod 3$:
$$ (2, 0, 1, 1) \\ (1, 0, 2, 2) \\ (2, 2, 0, 0) \\ (0, 2, 2, 2) $$
and the data points $(w,x,y,z) \bmod 5$: $$ (3, 0, 1, 2) \\ (3, 0, 2, 1) \\ (2, 0, 3, 4) \\ (2, 0, 4, 3) \\ (1, 2, 0, 4) \\ (0, 2, 1, 2) \\ (0, 2, 2, 1) \\ (1, 2, 4, 0) \\ (3, 4, 0, 4) \\ (4, 4, 2, 3) \\ (4, 4, 3, 2) \\ (3, 4, 4, 0) \\ $$ and we wanted to compute $(w,x,y,z) \pmod {3 \times 5}$.
One approach is to directly compute $(w,x,y,z) \pmod {3 \times 5}$ using the point-wise application of the Chinese Remainder Theorem to the cross-product.
We note that $(x,y,z)$ are all distinct. So, we could formulate $w(x,y,z)$ as a multivariate polynomial evaluated at those coordinates.
I am thinking of using the method outlined in this MSE answer for the interpolation.
Lagrange interpolation of multivariate polynomials, URL (version: 2020-08-11): https://math.stackexchange.com/q/3787199
specifically A Simple Expression for Multivariate Lagrange Interpolation by Kamron Saniee
Question: If we are just interested in a multivariate polynomial $w(x,y,z) \pmod {3 \times 5}$ and not the enumeration of the datapoints, could we do interpolation of $w(x,y,z) \pmod {3}$ and $w(x,y,z) \pmod {5}$ and then write down $w(x,y,z) \pmod{3 \times 5}$ in closed form using Chinese Remainder Theorem?