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A sufficient condition for a hypersurface in a polydisc U" to be the zero set of an HX(U") function is proved. This strengthens a result of Zarantonello and generalizes a result of Rudin. Using this result and a result of Andreotti and Stoll, a partial extension of Alexander's theorem on extension of bounded holomorphic functions from a hypersurface of U" to U" is obtained. Finally, a generalization of Cima's extension theorem for Hp functions is given.doi:10.2307/2041122 fatcat:zy5ee3re3rh6rc7777gh5vd4xi