On the complexity of invariant polynomials under the action of finite reflection groups

03/08/2022

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by   Thi Xuan Vu, et al.

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Let 𝕂[x_1, …, x_n] be a multivariate polynomial ring over a field 𝕂. Let (u_1, …, u_n) be a sequence of n algebraically independent elements in 𝕂[x_1, …, x_n]. Given a polynomial f in 𝕂[u_1, …, u_n], a subring of 𝕂[x_1, …, x_n] generated by the u_i's, we are interested infinding the unique polynomial f_ new in 𝕂[e_1,…, e_n], where e_1, …, e_n are new variables, such that f_new(u_1, …, u_n) = f(x_1, …, x_n). We provide an algorithm and analyze its arithmetic complexity to compute f_new knowing f and (u_1, …, u_n).