I have never developed sufficient knowledge in algebraic geometry but I ran into an apparently easy problem, so I apologise in advance if my question sounds naive. Suppose to have a degree $2$ homogeneous (non-trivial) polynomial $p(\mathbf{x})=\sum\limits_{j=1}^n\sum\limits_{i=1}^ja_{ij}x_ix_j\in\mathbb{F}_2[\mathbf{x}]$ of $n$ variables, with $\mathbb{F}_2=\mathbb{Z}/2\mathbb{Z}=\{0,1\}$. If $n\geq3$, we can use Chevalley-Warning theorem to conclude that the zero locus of $p$ must have even cardinality, whence odd cardinality of the zero locus of $p$ may occur only in the cases $n=1,2$ (for $n=1$, $p(x)=x^2$ does the job while for $n=2$ we have four possibilities, $p(x)=xy,x(x+y),(x+y)y,x^2+xy+y^2$). Do you think that Chevalley-Warning theorem can be avoided in this particular case, to prove that for $n\geq3$ the zero locus has even cardinality?
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1$\begingroup$ A counting argument not unlike that in the proof of Chevalley-Warning is simple over $\Bbb{F}_2$: The sum of values of a monomial $x_{i_1}x_{i_2}\cdots x_{i_k}$ of degree $k$ in $n$ variables, $n>k$ so choose any $k$ out of the $n$ to appear as factors, obviously vanishes, because there is duplication (the value of a missing variable can be chosen freely). Consequently the same holds for any sum of such monomials, implying the claim. $\endgroup$– Jyrki LahtonenCommented Mar 9 at 4:53
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1$\begingroup$ A bit more can be said. In coding-theory we often use the fact that a polynomial (homogeneous or not) of degree $r$ in $\Bbb{F}_2[x_1,x_2,\ldots,x_m]$ has at least $2^{m-r}$ zeros and at least $2^{m-r}$ non-zeros. Furthermore, either extreme is achieved only when the exceptional locus is a coset of an $r$-dimensional subspace. Look up Reed-Muller codes for more. $\endgroup$– Jyrki LahtonenCommented Mar 9 at 4:59
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