Regular sequences of power sums and complete symmetric polynomials

In this article, we carry out the investigation for regular sequences of symmetric polynomials in the polynomial ring in three and four variable. Any two power sum element in C[x_1, x_2, . . . , x_n] for n ≥ 3 always form a regular sequence and we state the conjecture when p_a, p_b, p_c for given positive integers a < b < c forms a regular sequence in C[x_1, x_2, x_3, x_4 ].We also provide evidence for this conjecture by proving it in special instances. We also prove that any sequence of power sums of the form p_a, p_a+1 , . . ., p_a+m−1 , p_b with m < n − 1 forms a regular sequence in C[x_1, x_2, . . . , x_n ]. We also provide a partial evidence in support of conjecture’s given by Conca, Krattenthaler and Watanble in [1] on regular sequences of symmetric polynomials.