I know the definition of a generalised Reed-Solomon Code:
$$ GRS_{n,k} (\alpha, v) = \{(v_0 f(\alpha_0), ..., v_{n-1}f(\alpha_{n-1}) ): deg f < k \}.$$
I know also that the product of two GRS codes is a GRS-code. But what is $n', k', \alpha', v'$ if $GRS_{n,k} (\alpha, v) * GRS_{n,k} (\alpha, v) = GRS_{n',k'} (\alpha', v')$?
A codeword in the product would have the following form $$ c = (v_1 f(\alpha_1)v_1g(\alpha_1), v_2 f(\alpha_2)v_2g(\alpha_2), \dots, v_n f(\alpha_n)v_n g(\alpha_n))\,. $$ So, $n' = n$, and as the product of two polynomials of degree at most $k$ can be a polynomial of degree at most $2k$ inclusive, $k' = 2k$. The code locators remain intact $\alpha' = \alpha$ and the column multipliers will be squared
$$ v' = (v_1^2, v_2^2, \dots, v_n^2)\,. $$