Intereting Posts

Explain a surprisingly simple optimization result
Is there a formula for $(1+i)^n+(1-i)^n$?
Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$
Why is $\pi$ = 3.14… instead of 6.28…?
Showing that $\int_0^\infty x^{-x} \mathrm{d}x \leq 2$.
In what sense is a tesseract (shown) 4-dimensional?
Factoring large integers without a cluster
Wikipedia Proof of Skolem-Noether Theorem
Can I get a PhD in Stochastic Analysis given this limited background?
Diophantine equation involving factorial …
Conditional expectation to de maximum $E(X_1\mid X_{(n)})$
A question regarding the Continuum Hypothesis (Revised)
Hamiltonian and non-Hamiltonian connected graph using the same degree sequence
Injectivity radius of Exponential and curvature
What's special about the greatest common divisor of a + b and a – b?

I’d like some help making this argument complete and rigorous (if it’s correct – if not, help with that would be nice).

Here $k$ is a field.

Let $A_1,\ldots,A_n \subseteq k$ be infinite subsets. Then any polynomial in $k[x_1,\ldots,x_n]$ that vanishes on $A_1\times\cdots\times A_n\subseteq k^n$ must be $0$ (as a polynomial).

This is what I have …

For the case $n=1$, a non-constant polynomial can only have as many roots as its degree, and in particular, it must have a finite number of roots. The only polynomial in one variable that has an infinite number of roots is $0$, so if a polynomial in $k[x_1,\ldots,x_n]$ vanishes on an infinite subset then it must be $0$.

For the inductive step, suppose the proposition is true for less than $n$ subsets and variables. Let $p\in k[x_1,\ldots,x_n]$ vanish on $A_1\times\cdots\times A_n$. Fix $x_n$ as some $a\in A_n$, and we have a polynomial in $n-1$ variables that vanishes on the set $A_1\times\cdots\times A_{n-1}$, so by the inductive hypothesis it must be identically $0$. (Now it gets sketchy). Since this is true for any of the infinite values in $A_n$, and , $p$ must be $0$.

- Hilbert's Original Proof of the Nullstellensatz
- For $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$ .
- Applications of $Ext^n$ in algebraic geometry
- Does there exist a regular map $\mathbb{A}^1\to\mathbb{P}^1$ which is surjective?
- What is the difference between Hom and Sheaf Hom?
- Help understanding Algebraic Geometry
- From the residue field at a point to a scheme
- Configuration scheme of $n$ points
- Transforming solvable equations to the de Moivre analogues
- Separatedness of a scheme of finite type over a field

Answered satisfactorily in the comments.

- 100 coin flips, expect to see 7 heads in a row
- If $a+b=1$ so $a^{4b^2}+b^{4a^2}\leq1$
- Solve the following non-homogeneous recurrence relation:
- Rating system incorporating experience
- $A \in M_3(\mathbb Z)$ be such that $\det(A)=1$ ; then what is the maximum possible number of entries of $A$ that are even ?
- If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal
- Sum of squared quadratic non-residues
- Integral Of $\frac{x^4+2x+4}{x^4-1}$
- Zero sections of any smooth vector bundle is smooth?
- Is the number 0.2343434343434.. rational?
- Subgroup generated by $2$ and $7$ in $(\mathbb Z,+)$
- Gradient of a Scalar Field is Perpendicular to Equipotential Surface
- Estimate of $n$th prime
- A problem about the twisted cubic
- What are the consequences if Axiom of Infinity is negated?