5

I have a homework problem where I have to show that if $f$ and $g$ are polynomials with no common factor in $K[x, y]$, where $K$ is a field, there are only finitely many elements $(a, b)\in K^2$ such that $f(a, b)=g(a, b)=0$.

My approach involves looking at $f$ and $g$ as polynomials in $y$ whose coefficients are polynomials in $x$. Then I can write the Sylvester matrix of $f$ and $g$, which is a matrix whose entries are polynomials in $x$. I've shown that if $f$ and $g$ have a common root $(a, b)$, then this matrix's determinant evaluated at $x=a$ would be $0$. Thus, if this determinant, viewed as a polynomial over $x$ is not the zero polynomial, it only has finitely many roots, and thus there are only finitely many values of $x$ that can be in a common root, and I'd be done.

Now I have to deal with the case where this determinant is identically the zero polynomial. If so, that means, that for every $a\in K$, $f(a, y)$ and $g(a, y)\in K[y]$ have a common root. I'm not quite sure how to get from this to my desired contradiction...

Something that may or may not be helpful is that I was able to show, from Gauss' Lemma and the fact that $K(x)[y]$ is a Principal Ideal Domain ($K(x)$ is the field of rational functions of $x$), that if there's a $b\in K$ such that both $f(x, b)$ and $g(x, b)\in K[x]$ are both the zero polynomial, then $f$ and $g$ have a common factor in the Unique Factorization Domain $K[x, y]$, a contradiction. Does this result help me prove that if the Sylvester determinant is zero, then $f$ and $g$ have a common factor?

I know one way of doing this is to prove that the Sylvester Determinant is equal to the resultant, but that proof seems rather long, requiring me to consider the algebraic closure of a field of fractions, and I don't think I need the full force of the result...is there an easier way to prove the weaker result that if the determinant is $0$, then $f$ and $g$ have a common root/factor.

Please don't give me full solutions; I just want some hints in the right direction so I can finish the problem on my own.

glS
  • 6,818
Nishant
  • 9,155

1 Answers1

1

Question: "Please don't give me full solutions; I just want some hints in the right direction so I can finish the problem on my own."

Hint: If two homogeneous polynomials $F(x,y,z), G(x,y,z)$ have no common factor, it follows by a classical results that the intersection $Z(F) \cap Z(G) \subseteq \mathbb{P}^2_k$ is a finite set of points $m$ with $m \leq deg(F)deg(G)$

hm2020
  • 1