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There exist integers A1,A2 - IMO LongList 1992 THA2

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September 2, 2010
algebrapolynomialnumber theoryrelatively primefunctional equationIMO ShortlistIMO Longlist

Problem Statement

Let P1(x,y)P_1(x, y) and P2(x,y)P_2(x, y) be two relatively prime polynomials with complex coefficients. Let Q(x,y)Q(x, y) and R(x,y)R(x, y) be polynomials with complex coefficients and each of degree not exceeding dd. Prove that there exist two integers A1,A2A_1, A_2 not simultaneously zero with ∣Ai∣≤d+1 (i=1,2)|A_i| \leq d + 1 \ (i = 1, 2) and such that the polynomial A1P1(x,y)+A2P2(x,y)A_1P_1(x, y) + A_2P_2(x, y) is coprime to Q(x,y)Q(x, y) and R(x,y).R(x, y).
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