Semidefinite representations for finite varieties

We present a concise semidefinite formulation for the problem of minimizing a polynomial over a semi-algebraic set defined by polynomial equalities and inequalities. When the polynomial equalities define a radical ideal I with a finite variety, this representation involves combinatorial moment matrices, indexed by a basis of R[x1, . . . , xn]/I. The arguments are elementary and extend known facts for the grid case including 0/1 and ±1 programming. Semidefinite approximations are obtained by considering truncated combinatorial moment matrices; rank conditions are given that ensure that the approximation solves the original problem at optimality.