Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 109 / 2

For a sequence $J$ in ${\mathbb{F}}_p^2$, write $(r\mid J)$ for the number of $r$-term subsequences whose sum is zero. We use the following consequence of the Chevalley-Warning theorem: if $|J|=3p-1$, thenIndeed, with one variable $x_i$ for each term $(a_i,b_i)$, apply Chevalley--Warning toTheir degree sum is $3(p-1)<3p-1$. A common zero has support of size $0,p$, or $2p$ modulo $p$, and each fixed support contributes $(p-1{)}^{|S|}$ assignments. Reducing modulo $p$ gives the displayed congruence.

Now let the given $3p$ terms have total sum zero. If no $p$ of them summed to zero, delete any one term and apply the congruence to the remaining $3p-1$ terms. It givesso those remaining terms contain a zero-sum $2p$-subsequence. Its complement in the original $3p$ terms has size $p$ and sum zero, contradicting the assumption. Therefore the required $p$ terms exist.

We use the coefficient form of the Combinatorial Nullstellensatz: if a polynomial of degree ${\sum }_i{t}_i$ has nonzero coefficient at ${\prod }_i{x}_i^{t_i}$, then it cannot vanish on every product set ${\prod }_i{S}_i$ with $|S_i|>t_i$.

Write the terms as $(a_i,b_i)$ and, over ${\mathbb{F}}_p$, putandConsiderA direct multinomial-coefficient calculation, using Wilson's theorem, shows that the coefficient of ${\prod }_{i=1}^{4p-2}{x}_i$ in $F$ is nonzero. The Combinatorial Nullstellensatz with every $S_i=\{0,1\}$ therefore gives an indicator vector $x$ for which $F(x)\ne 0$.

The zero indicator does not work. The first two factors force the selected vectors to have both coordinate sums zero. If the support size is not divisible by $p$, Fermat's little theorem makes $1-S^{p-1}=0$, while the final product also vanishes; hence the support has size $p$, $2p$, or $3p$. Size $p$ proves the claim, and size $3p$ reduces to part (a). For size $2p$,so the bracket vanishes, again contradicting $F(x)\ne 0$. Only the size-$p$ and reducible size-$3p$ cases remain, and either yields the required zero-sum $p$-subsequence.