On extension of multilinear operators and homogeneous polynomials in vector lattices

We establish the existence of a simultaneous extension from a majorizing sublattice in the classes of regular multilinear operators and regular homogeneous polynomials on vector lattices. By simultaneous extension from a sublattice we mean a right inverse of the restriction operator to this sublattice which is an order continuous lattice homomorphism. The main theorems generalize some earlier results for orthogonally additive polynomials and bilinear operators. The proofs base on linearization by Fremlin's tensor product and the existence of a right inverse of an order continuous operator with Levy and Maharam property.