MHB A Conjecture About Polynomials in Two Variables

caffeinemachine
Gold Member
MHB
Messages
799
Reaction score
15
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?
 
Physics news on Phys.org
caffeinemachine said:
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?]
 
Opalg said:
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.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
Back
Top