Take the 2-minute tour ×  Consider the $\mathbb Z$-module that
consists of the polynomials in $\mathbb Z[x,y]$ that are
homogeneous polynomials of degree $d$ in the indeterminates $x$
and $y$ (homogeneous meaning that all terms of the form $c_{ij}
x^i y^j$ are such that $i+j = d$, where $d$ is the degree of the
polynomial). One can see that there is a unique way of writing
an homogeneous polynomial of degree $d$ in the form $$
\sum_{j=0}^d c_k y^{d-k} \prod_{i=0}^{k-1} (x-iy) $$ because
there is precisely one term where $x$ is at the $k^{th}$ power
for any $k$ in the range $[0,d]$, hence we can compute
coefficients. Therefore, the polynomials $y^{d-k} \prod (x-iy)$
form a basis of the $\mathbb Z$-module.   Is it possible to
write any homogenous polynomial of degree $d$ in the form $$
\sum_{k=0}^d c_k \left( \prod_{i=0}^{d-k-1} (y-ipx) \right)
\left( \prod_{i=0}^{k-1} (x-iy) \right) $$ where $p$ is a prime
number? (The larger context is a number theory context, thus the
prime is the thing I need. The fact that