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|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil} and P(x, y) = P(y,- x -y)

Source: RMM Shortlish 2016 A2

July 4, 2019
ceiling functionalgebrapolynomialprime

Problem Statement

Let p>3p > 3 be a prime number, and let FpF_p denote the (fi nite) set of residue classes modulo pp. Let SdS_d denote the set of 22-variable polynomials P(x,y)P(x, y) with coefficients in FpF_p, total degree ≤d\le d, and satisfying P(x,y)=P(y,−x−y)P(x, y) = P(y,- x -y). Show that ∣Sd∣=p⌈(d+1)(d+2)/6⌉|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil}. The total degree of a 22-variable polynomial P(x,y)P(x, y) is the largest value of i+ji + j among monomials xiyjx^iy^j appearing in PP.
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