Relaxation results for functions depending on polynomials changing sign on rank-one matrices

In this paper, we are interested in computing the different convex envelopes of functions depending on polynomials, especially those having it is main part change sign on rank-one matrices. Our main result applies to functions of the type W ( F ) = φ ( P ( F ) ) , W ( F ) = φ ( P ( F ) ) + f ( det F ) or W ( F ) = φ ( P ( F ) ) + g ( adj n F ) defined on the space of matrices, where φ, f : R → R and g : R 3 → R are three continuous functions, and P = P 0 + P 1 + ⋯ + P d is a polynomial such that P d has the property of changing sign on rank-one matrices. Then the polyconvex, quasi-convex and rank-one convex envelopes of W are equal.