Let $K$ be a field. I want to store a homogeneous quadratic map $Q : K^n \to K^m$, given by $m$ homogeneous quadratic polynomials in $n$ variables. To store this in memory we need to store the $m \times \frac{n(n+1)}{2}$ coefficients of the system. However, I don't care about what basis for $K^n$ this system is expressed in, and I want to exploit this in order to save some memory.
Given a quadratic polynomial, I know that it is possible to pick a basis such that the polynomial is diagonal (i.e. of the form $\sum_i a_i X_i^2$). So we can pick a basis such that the first polynomial is diagonal and store only its diagonal terms, so in total we need to store $n + (m-1)\times\frac{n(n+1)}{2}$ coefficients.
With this trick we save $\frac{n(n-1)}{2}$ coefficients, but I think it should be possible to do even better.
Is it possible to do better? What is the best way to do this? Thanks in advance!