TSTST 2026 Problem 3

Suppose $f_1, f_2, \ldots, f_{1000} \in \mathbb{C}[x]$ are single-variable polynomials with complex coefficients such that there do not exist distinct $i, j \in \{1, 2, \ldots, 1000\}$ and $a \in \mathbb{C}$ with $f_i = a f_j$. Given that the degree of $f_1$ is 10, find the minimum possible total degree of

$$\sum_{k=1}^{1000} f_k(x_1) f_k(x_2) \cdots f_k(x_{2026}).$$

Note: For example, the total degree of the polynomial $g(x, y) = 20x^3y^4 + 26x^6$ is 7 because $7 = 3 + 4 > 6$.