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VJIMC 2019 P2

Source: VJIMC 2019

March 29, 2019
algebrapolynomialreal analysisVJIMC2019VJIMCVojtech JarnikAnnual Vojtech Jarnic

Problem Statement

A triplet of polynomials u,v,w∈R[x,y,z]u,v,w \in \mathbb{R}[x,y,z] is called smart if there exists polynomials P,Q,R∈R[x,y,z]P,Q,R\in \mathbb{R}[x,y,z] such that the following polynomial identity holds :u2019P+v2019Q+w2019R=2019u^{2019}P +v^{2019 }Q+w^{2019} R=2019
a) Is the triplet of polynomials u=x+2y+3,        v=y+z+2,          w=x+y+zu=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y+z smart? b) Is the triplet of polynomials u=x+2y+3,        v=y+z+2,          w=x+y−zu=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y-z smart?
Proposed by Arturas Dubickas (Vilnius University).
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