For an n-tuple of compact operators T=(T₁,...,T_n) on a Hilbert space H we consider a notion of joint spectrum of T, denoted by Σ(T), which consists of points z=(z₁,...,z_n) in Cⁿ such that I+z₁T₁+...+z_nT_n is not invertible, where I is the identity operator on H. Using the theory of determinants for certain Fredholm operators we show that Σ(T) is always an analytic set of codimension 1 in Cⁿ. This result is in fact a special case of a multivariable version of the analytic Fredholm theorem.