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A Conjecture About Polynomials in Two Variables

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
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

MHB Oldtimer
Staff member
Feb 7, 2012
2,701
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

Well-known member
MHB Math Scholar
Mar 10, 2012
834
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.