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(detF) or W(F)=φ(P(F))+g(adjnF) defined on the space of matrices, where φ, f:R→R and g:R3→R are three continuous functions, and P=P0+P1+⋯+Pd is a polynomial such that Pd has the property of changing sign on rank-one matrices. Then the polyconvex, quasi-convex and rank-one convex envelopes of W are equal.