On two classes of regular sequences (Q262657)

In the paper under review, the author tries to construct some examples of regular sequences in the polynomial ring over the field of complex numbers. The author mentions that there are two conjectures about these regular sequences; one of them is the famous EGH conjecture. Let \(p_a(n)\) denote the polynomial \(x_1^a+x_2^a+\dots +x_n^a\), for every two positive integers \(a,n\). NEWLINENEWLINENEWLINE In this paper, it is shown that if \(\gcd(a,d)=1\), then \(p_a(n),p_{a+d}(n),\dots, p_{a+(n-1)d}(n)\) is a regular sequence. For the homogeneous polynomials \(f=\lambda_1x_1^d+\lambda_2x_1^{d-1}x_2+\dots + \lambda_rx_n^d\) and \(g=\nu_1x_1^d+\nu_2x_1^{d-1}x_2+\dots +\nu_r x_n^d\), the author defines the distance between \(f\) and \(g\) to be \(d(f,g):=\sum_{i=1}^r |\lambda_i-\nu_i|\). Given homogeneous polynomials \(f_1,f_2,\dots, f_n\) of degrees \(1 \leq a_1 \leq a_2 \leq\dots \leq a_n\) in \(\mathbb{C}[x_1,x_2,\dots, x_n]\), it has been proved that if \(d(f_i,x_i^{a_i})<1\), then \(f_1,f_2,\dots, f_n\) is a regular sequence.