The Role of Non-Negative Polynomials For Rank-One Convexity and Quasi Convexity

We stress the relationship between the non-negativeness of polynomials and quasi convexity and rank-one convexity. In particular, we translate the celebrated theorem of Hilbert ([3]) about non-negativeness of polynomials and sums of squares, into a test for rank-one convex functions defined on 2 × 2-matrices. Even if the density for an integral functional is a fourth-degree, homogeneous polynomial, quasi convexity cannot be reduced to the non-negativeness of polynomials of a fixed, finite number of variables.