MathDB

Given an integer

n \geqslant 3

the polynomial

f(x_1, \ldots, x_n)

with integer coefficients is called good if

f(0,\ldots, 0) = 0

and

f(x_1, \ldots, x_n)=f(x_{\pi_1}, \ldots, x_{\pi_n}),

for any permutation of

\pi

of the numbers

1,\ldots, n

. Denote by

\mathcal{J}

the set of polynomials of the form

p_1q_1+\cdots+p_mq_m,

where

m

is a positive integer and

q_1,\ldots , q_m

are polynomials with integer coefficients, and

p_1,\ldots , p_m

are good polynomials. Find the smallest natural number

D{}

such that each monomial of degree

D{}

lies in the set

\mathcal{J}

Problem Statement