How many n-tuple to genereate zero from some random several variable equation that use constant power and a variable as the base?

Question

Define algebraic number tuple as the aphabetical order sequence of variables that use in the equation. How many algebraic number n-tuple (x,y,z,...) are able to genereate zero to input into a several variable diophaine equation that use integer power?

For example: $x^6y^9z^2+yx+5$=0 is such a equation

related question: How does Hilbert's Nullstellensatz generalize the "fundamental theorem of algebra"?

Answer 1

I'm going to first interpret your question:

"Can you describe the set of $n$-tuples of real numbers $(a_1, a_2, \dots , a_n)$ such that there's some polynomial with rational coefficients $f \in \Bbb Q[x_1, \dots , x_n]$ with $f(a_1, \dots , a_n) = 0$?"

For the $n=1$ case, you get the algebraic numbers. These are precisely the real numbers that are the roots of some polynomial with rational coefficients. If I'm to interpret "how many" as the cardinality of this set, the algebraic numbers are countable.

For $n \geq 2$ you (essentially) recover the concept of algebraic independence of real numbers over the rationals. I'll focus on the $n=2$ case.

• If one of $x_1$ or $x_2$ is an algebraic number, then $(x_1,x_2)$ is in your set. Proof: let's say $x_1$ is algebraic; that means that there's some polynomial $p(x)$ with rational coefficients such that $p(x_1)=0$. Then define $p'(x,y) = p(x)$. Then $p'(x_1,x_2) = p(x_1) = 0$, so $(x_1, x_2)$ is in your set.
• If neither of $x_1$ or $x_2$ is an algebraic number, then it's anybody's guess whether or not $(x_1,x_2)$ is in your set. As in the wikipedia article, $(\sqrt{\pi}, 2\pi+1)$ is in your set, since it satisfies $2x^2-y+1$. But it's an open problem as to whether or not $(e,\pi)$ is in your set, for instance.
• Because of the first bullet point, there are uncountably many $n$-tuples in your set. If you think it's "cheating" that I allow polynomials with only one of the two variables, let's say the polynomials have to have two variables; then this set is still uncountable: consider the solutions to $y-x^2=0$, for instance.

Questions about your set lie firmly in the field of transcendental number theory. You might be interested in the various things linked in that Wikipedia article.