Could anyone point me to a reference showing that the zero set of a polynomial in $n \ge 2$ variables has Lebesgue measure zero? I wonder if there are pathological examples, and some conditions needed for this.
Here is a sketch of an argument I have: Let $p(x,y) \equiv p(x,y_1,\dots,y_n)$ be a polynomial of degree $d$ (with real coefficients say) in $n+1$ variables. Assume that the polynomial is not identically zero. Let $A = \{(x,y) \in \mathbb R^{n+1} :\; p(x,y) = 0\}$. I would like to argue that this set has Lebesgue measure zero, by induction on $n$.
Consider the set of all $y$ for which $x \mapsto p(x,y)$ is identically zero, i.e. $B = \{y \in \mathbb R ^n:\; p(\cdot,y) = 0\}$. Since the coefficients of $x$ in $x \mapsto p(x,y)$ are polynomials in $y$, $B$ can be written as the zero set of a system of polynomials in $y$. By induction assumption, $B$ should have measure zero. Now $A \subset ([0,1]\times B) \cup (A \cap ([0,1]\times B^c))$ where \begin{align} A_2 := A \cap ([0,1]\times B^c) &= \{(x,y) \in \mathbb R^{n+1} : y\in B^c,\; \; p(x,y) = 0\}\\ &= \{(x,y) \in \mathbb R^{n+1} : y\in B^c,\; \; x\in Z_y\} \end{align} where $Z_y$ is the zero set of $p(\cdot,y)$ which is a finite set with at most $d$ elements. Since $A$ and $B$ are measurable, $A_2$ is measurable (and $Z_y$ is certainly measurable), and we can use disintegration theorem (?) to get $$ \mu_{n+1}(A_2) = \int_{y \in B^c} \mu_n(Z_y) dy = 0 $$ where $\mu_n$ is the $n$-dimensional Lebesgue measure. Similarly by Fubini $\mu_{n+1}([0,1]\times B) = 0$. It follows that $\mu_{n+1}(A) =0$.
Is there something wrong with the above argument?
EDIT: I guess one has to rule out cases like this $p(x,y) = x^2 + ((y+1)^2 - y^2-2y-1)x$ where a coefficient is identically zero for my argument to go through. Can we say that for every polynomial there is another one where these trivial identities have been removed, and they are the same as functions (or at least have the same zero sets)? Unfortunately, I don't know/remember enough algebraic geometry to know if what I am saying is something obviously true or obviously false or even phrased correctly?
