# Irreducible polynomial equations from equations of algebraic functions in two variables?

Mathematics Asked by IV_ on November 27, 2020

Let $$A$$ be an arbitrary algebraic function in dependence of the two variables $$x$$ and $$y$$.
Let’s consider the equation $$A(x,y)=0$$ (equation 1) involving $$x$$ and $$y$$. For which kinds of equation 1 can we decide if there is an irreducible polynomial equation whose solution set contains the solution set of equation (1)?

I already know that we can multiply both sides of the equation by the denominators of both sides to get an equivalent or non-equivalent equation. And the solution set of this new equation does contain the equivalent solution set. At least in the one-variable case, the equivalent solution set yields an equivalent polynomial equation.

I also see: We can first separate the denominators and the irreducible factors.

The following is a simple first case of corresponding radical equations.

Let
$$A_0inoverline{mathbb{Q}}[x,y]$$ algebraic,
$$Pinoverline{mathbb{Q}}[t]setminusoverline{mathbb{Q}}$$,
$$P_1inoverline{mathbb{Q}}[x,y]setminus(overline{mathbb{Q}}[x]cupoverline{mathbb{Q}}[y])$$,
$$rinmathbb{Q}setminusmathbb{Z}$$
so that $$P((P_1(x,y))^r)$$ and $$A_0(x,y)$$ are coprime,
and let $$P^{-1}$$ denote a suitable partial inverse of $$P$$.

$$frac{P((P_1(x,y))^r)}{A_0(x,y)}=0$$

$$P((P_1(x,y))^r)=0$$

$$(P_1(x,y))^r=P^{-1}(0)$$

$$P_1(x,y)=(P^{-1}(0))^frac{1}{r}$$

$$P_1(x,y)-(P^{-1}(0))^frac{1}{r}=0$$

The equation is irreducible if $$P$$ is irreducible and $$P_1(x,y)-(P^{-1}(0))^frac{1}{r}$$ is irreducible for all suitable partial inverses $$P^{-1}$$.

Correct answer by IV_ on November 27, 2020

There may not be. Let A(x,y) = x/y.
A(x,y) = 0 iff x = 0 and y /= 0.
No polynomial can have that property.

Answered by William Elliot on November 27, 2020