Математичнi Студiї. Т.50, №1 Matematychni Studii. V.50, No.1 УДК 517.98 T. V. Vasylyshyn SYMMETRIC POLYNOMIALS ON THE CARTESIAN POWER OF L p ON THE SEMI-AXIS T. V. Vasylyshyn. Symmetric polynomials on the Cartesian power of L p on the semi-axis, Mat. Stud. 50 (2018), 93-104. The paper deals with polynomials in the complex Banach space (L p [0, +∞)) n , which are the nth Cartesian power of the complex Banach space of Lebesgue measurable integrable in a power p complex-valued functions on [0,

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... where 1 ≤ p < +∞. It is proved that if p is an integer, then every continuous symmetric polynomial on (L p [0, +∞)) n can be uniquely represented as an algebraic combination of some "elementary" p-homogeneous symmetric polynomials. It is also proved that if p is not an integer, then every continuous symmetric polynomial on (L p [0, +∞)) n is constant. Results of the paper can be used for investigations of algebras of symmetric continuous polynomials and of symmetric analytic functions on (L p [0, +∞)) n .