Note on the Algebraic Precision of Reducing-Dimensionality Expansion

In formula (1) we can choose an auxilliary polynomial P(x) from the class of polyncmials K_m={P_m(x)|P_m(x)∈P_m and the coefficient of the term x_1~(m_3)x_2~(m_2)…x_n~(m_n) is 1, m_1+m_2+…+m_n=m}, in which P_m denotes the space of all polynomials of degree≤m. Theorem For any bounded region in R~n and any given positive integer m, there exists a class of auxilliary polynomials P(x)∈K_m in (1) such that the reducingdimensionality expansion (1) is of the highest algebraic precision 2m-1.