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 $\mathbb{C}[x_1,x_2,\dots,x_n]$ for $n \geq 3$ 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 $\mathbb{C}[x_1,x_2,x_3,x_4]$. 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 $p_{a}, p_{a+1},\dots, p_{a+ m-1},p_b$ with $m <n-1$ forms a regular sequence in $\mathbb{C}[x_1,x_2,\dots,x_n]$. We also provide a partial evidence in support of conjecture's given by Conca, Krattenthaler and Watanabe on regular sequences of symmetric polynomials.