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common zeros of polynomials

Source: Komal A.718

March 13, 2018
algebrapolynomial

Problem Statement

Let R[x,y]\mathbb{R}[x,y] denote the set of two-variable polynomials with real coefficients. We say that the pair (a,b)(a,b) is a zero of the polynomial f∈R[x,y]f \in \mathbb{R}[x,y] if f(a,b)=0f(a,b)=0.
If polynomials p,q∈R[x,y]p,q \in \mathbb{R}[x,y] have infinitely many common zeros, does it follow that there exists a non-constant polynomial r∈R[x,y]r \in \mathbb{R}[x,y] which is a factor of both pp and qq?
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