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Theoretical analysis demonstrates that conjugation-closed subsets containing zero form exceptional sets for multivariable transcendental functions, expanding classical single-variable results.
In 2020, Marques and Moreira proved that every subset of [Formula: see text], which is closed under complex conjugation and contains [Formula: see text], is the exceptional set of uncountably many transcendental analytic functions with integer coefficients. In this paper, we extend this result to transcendental analytic functions in several variables. In particular, we show that every subset of [Formula: see text] m contained in the unit polydisc [Formula: see text], closed under complex conjugation and containing the zero vector, is the exceptional set of uncountably many transcendental analytic functions in several variables with integer coefficients.
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Lelis et al. (2026) studied this question.
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