are there any bounds known for approximating a genuine multidimensional polynomial function with a sum one-dimensional polynomials over the independent variables?

In the twodimensional case the expression to be minimized would be: $$\left\| \sum_{i=0}^m\sum_{j=0}^nc_{ij}x^iy^j\ - \left(\sum_{k=0}^{m+n} a_kx^k+b_ky^k\right) \right\|$$

I am mainly interested in the standard cases of the $L_p$ norms (i.e. for $p=\infty,2,1$ in the order of preference), but results related to other error measures are also appreciated.