A representation theorem is obtained for generalized infrapolynomials on an arbitrary complex point set. Some results on the location of zeros are also presented. Let P denote the class of polynomials In c zi with complex coefficients considered as mappings of the complex z-plane C z into itself. Let __i IS denote a fixed set of s linearly independent linear functionals on Pn, and A=A1, A 2, ., As be a fixed s-tuple of complex numbers. Then Pn(A) will represent the class of polynomials p(z) in Pn such that Sip = Ai, i = 1, 2, s. Further, let E denote a compact subset of C containing at least n s + 2 points. As in |l] we make the following Definition. p(z) E Pn(A) is called an infrapolynomial on E with respect to Pn(A) if p(z) has on E no underpolynomials in P n(A); i.e., if there exists no polynomial q(z) in Pn(A) such that (1) q(z)j < Ip(z)I on E nIz; p(z) 7 07, (2) q(z) = 0 on E nI z; p(z) = 07. A polynomial q(z) E Pn(A) such that (3) 1q(z)j < Ip(z)I on E is called a weak underpolynomial of p(z) on E with respect to Pn(A). Let ez denote the functional which gives point evaluation at z, i.e. ezf = /W) Haar Assumption I. 12S I U 1 e is a linearly independent set of functionals on Pn for all choices of n s + 1 distinct points Zs+1 Zn + in E. Note 1. Recall that, by assumption, E contains at least n s + 2 points. Note that if E were to contain only n s + 1 points, Pn(A) would contain a unique infrapolynomial, vanishing of course at all the points of E. Received by the editors September 19, 1972 and, in revised form, February 3, 1974. AMS (MOS) subject classifications (1970). Primary 30A82; Secondary 30A08.