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A sufficient condition for a hypersurface in a polydisc U' to be the zero set of an H (Ul) 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.

Zero sets and extensions of bounded holomorphic functions in polydiscs