A Conjecture About Polynomials in Two Variables

[caffeinemachine]

Let $p(x,y)$ and $q(x,y)$ be two polynomials with coefficients in $\mathbb R$. Define $P=\{(a,b)\in\mathbb R^2 : p(a,b)=0\}$ and $Q=\{(a,b)\in \mathbb R^2:q(a,b)=0\}$. Now assume that there is a sequence of points $(x_n,y_n)$ in $\mathbb R^2$ such that:

1. $(x_n,y_n)\to (0,0)$.
2. $(x_n,y_n)\in P\cap Q$ for all $n$.

I conjecture that there is a continuous curve $\Gamma:[0,1)\to\mathbb R^2$ such that $\Gamma(0)=(0,0)$ and $\Gamma(t)\in P\cap Q$ for all $t>0$.

I am pretty sure that the above is true. But I have no ideas on how to go about proving it. If you draw a picture I think you will also be convinced that the above should be true. Can somebody help?

 

[Opalg]

Let $p(x,y)$ and $q(x,y)$ be two polynomials with coefficients in $\mathbb R$. Define $P=\{(a,b)\in\mathbb R^2 : p(a,b)=0\}$ and $Q=\{(a,b)\in \mathbb R^2:q(a,b)=0\}$. Now assume that there is a sequence of points $(x_n,y_n)$ in $\mathbb R^2$ such that:

1. $(x_n,y_n)\to (0,0)$.
2. $(x_n,y_n)\in P\cap Q$ for all $n$.

I conjecture that there is a continuous curve $\Gamma:[0,1)\to\mathbb R^2$ such that $\Gamma(0)=(0,0)$ and $\Gamma(t)\in P\cap Q$ for all $t>0$.

I am pretty sure that the above is true. But I have no ideas on how to go about proving it. If you draw a picture I think you will also be convinced that the above should be true. Can somebody help?

You have two algebraic curves $P$ and $Q$, and they have infinitely many points of intersection. It follows from Bézout's theorem that $p(x,y)$ and $q(x,y)$ have a non-constant polynomial greatest common divisor $r(x,y)$. I guess that the curve $\{(x,y):r(x,y)=0\}$ should somehow solve your problem.

[But I am not an algebraic geometer. Maybe someone else here has some expertise in that area?]

 

[caffeinemachine]

You have two algebraic curves $P$ and $Q$, and they have infinitely many points of intersection. It follows from Bézout's theorem that $p(x,y)$ and $q(x,y)$ have a non-constant polynomial greatest common divisor $r(x,y)$. I guess that the curve $\{(x,y):r(x,y)=0\}$ should somehow solve your problem.

[But I am not an algebraic geometer. Maybe someone else here has some expertise in that area?]

Thanks. Let me read a bit and get back.