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Minimal number of polynomials to write as SOS

Source: stems 2021 cat b/c p4

January 24, 2021
algebrapolynomial

Problem Statement

Let nn be a fixed positive integer. - Show that there exist real polynomials p1,p2,p3,⋯ ,pk∈R[x1,⋯ ,xn]p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n] such that (x1+x2+⋯+xn)2+p1(x1,⋯ ,xn)2+p2(x1,⋯ ,xn)2+⋯+pk(x1,⋯ ,xn)2=n(x12+x22+⋯+xn2)(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2) - Find the least natural number kk, depending on nn, such that the above polynomials p1,p2,⋯ ,pkp_1, p_2, \cdots, p_k exist.
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