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Integer Symmetric Polynomials

Source: Indian Statistical Institute (ISI) UGB 2023 P7

May 14, 2023
algebrapolynomialisiIndian Statistical Institute

Problem Statement

(a) Let n≥1n \geq 1 be an integer. Prove that Xn+Yn+ZnX^n+Y^n+Z^n can be written as a polynomial with integer coefficients in the variables α=X+Y+Z\alpha=X+Y+Z, β=XY+YZ+ZX\beta= XY+YZ+ZX and γ=XYZ\gamma = XYZ. (b) Let Gn=xnsin⁡(nA)+ynsin⁡(nB)+znsin⁡(nC)G_n=x^n \sin(nA)+y^n \sin(nB)+z^n \sin(nC), where x,y,z,A,B,Cx,y,z, A,B,C are real numbers such that A+B+CA+B+C is an integral multiple of π\pi. Using (a) or otherwise show that if G1=G2=0G_1=G_2=0, then Gn=0G_n=0 for all positive integers nn.
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