Generalized Hamming weights of affine Cartesian codes

Let F be any field and A1,…,Am be finite subsets of F. We determine the maximum number of common zeroes a linearly independent family of r polynomials of degree at most d of F[x1,…,xm] can have in A1×…×Am. In the case when F is a finite field, our results resolve the problem of determining the generalized Hamming weights of affine Cartesian codes. This is a generalization of the work of Heijnen and Pellikaan where these were determined for the generalized Reed-Muller codes. Finally, we determine the duals of affine Cartesian codes and compute their generalized Hamming weights as well.