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