Consider the given system of polynomial equations, where all the coefficients are in <mi mathvari

Jonathan Miles

Jonathan Miles

Answered question

2022-07-02

Consider the given system of polynomial equations, where all the coefficients are in C:
{ y n = P ( x ) Q ( x , y ) = 0
I would like to establish that either this system has solutions ( x , y ) C for all x C, or either has solutions for countably (or even better, finitely) many x C. I am not completely sure this is true but haven't found any counterexamples yet.
So far, I have found that if we have solutions for infinitely many x, then we have solutions for infinitely many y.

Answer & Explanation

lywiau63

lywiau63

Beginner2022-07-03Added 13 answers

You can determine the resultant R ( x ) = R e s y ( y n P ( x ) , Q ( x , y ) ) of both polynomials eliminating y. If it is the zero polynomial, then the polynomials have a common factor, which means that for any x you find at least one y so that ( x , y ) a solution (of the common factor and thus of the system).
If the resultant is not zero, then R ( x ) is a polynomial. Only at the roots of this polynomial you will then find at least one y so that ( x , y ) is a solution of the system.

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