Problem

Source: RMM Shortlish 2016 A2

Tags: ceiling function, algebra, polynomial, prime



Let $p > 3$ be a prime number, and let $F_p$ denote the (finite) set of residue classes modulo $p$. Let $S_d$ denote the set of $2$-variable polynomials $P(x, y)$ with coefficients in $F_p$, total degree $\le d$, and satisfying $P(x, y) = P(y,- x -y)$. Show that $$|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil}$$. The total degree of a $2$-variable polynomial $P(x, y)$ is the largest value of $i + j$ among monomials $x^iy^j$ appearing in $P$.