Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 145 / 5 / a / Solution

The Alon-Tarsi lemma says that if $\mathrm{deg}f\le d_1+\cdots +d_n$ and $A_i$ consists of $d_i+1$ distinct field elements, thenThis follows by applying univariate Lagrange interpolation successively in each variable.

If the displayed coefficient is nonzero, at least one summand has $f(a_1,\dots ,a_n)\ne 0$. This is the coefficient form of the Combinatorial Nullstellensatz.

Solved by gpt-5.6-sol high.