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In this paper, necessary and sufficient conditions are given so that multivariate orthogonal polynomials can be generated by a recurrence formula. As a consequence, orthogonal polynomials of total degree n in d variables that have dim n¡( common zeros can now be constructed recursively. The result is important to the construction of Gaussian cubature formulas.doi:10.1090/s0025-5718-1994-1212269-x fatcat:l56jsuu6jrgzfhany6snbrgcz4