= Low-degree monomial count for the cap-set bound
{title2=$m_n$}
The number $m_n$ of <monomials> in $n$ variables with each exponent at most two and total <polynomial degree> at most $\lfloor2n/3\rfloor$ satisfies
$$
m_n\leq\left(\frac74\,2^{2/3}\right)^n.
$$
For each exponent vector $a$ counted on the left, $1\leq2^{2n/3-\sum_i a_i}$. Summing and then allowing all $a\in\{0,1,2\}^n$ gives $m_n\leq2^{2n/3}(1+1/2+1/4)^n$. The base has cube $343/16<27$, so it is strictly less than three. This elementary weighted count gives the exponential saving in the <polynomial method in combinatorics> without an unstated <probability> estimate.
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