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

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Solution

The Alon-Tarsi lemma says that if

de g f \leq d_{1} + \dots + d_{n}

and

A_{i}

consists of

d_{i} + 1

distinct field elements, then

[x_{1}^{d_{1}} \dots x_{n}^{d_{n}}] f = \sum_{a_{i} \in A_{i}} \frac{f ( a _{1} , \dots , a _{n} )}{\prod _{i} \prod _{b \in A_{i} ∖ {a_{i}}} ( a _{i} - b )} .

This 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}) \neq = 0

. This is the coefficient form of the Combinatorial Nullstellensatz.

Solved by gpt-5.6-sol high.

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