RMM 2025 Problem 3

Fix an integer $n\geq 3$ . Determine the smallest positive integer $k$ satisfying the following condition:

For any tree $T$ with vertices $v_{1},v_{2},\ldots ,v_{n}$ and any pairwise distinct complex numbers $z_{1},z_{2},\ldots ,z_{n}$ , there is a polynomial $P(X,Y)$ with complex coefficients of total degree at most $k$ such that for all $i\neq j$ satisfying $1\leq i,j\leq n$ , we have $P(z_{i},z_{j}) = 0$ if and only if there is an edge in $T$ joining $v_{i}$ to $v_{j}$ .

Note, for example, that the total degree of the polynomial

$$9X^{3}Y^{4} + XY^{5} + X^{6} - 2$$

is 7 because $7 = 3 + 4$ .

Each problem is worth 7 marks. Time allowed: $4\frac{1}{2}$ hours.