Lower bounds on heights of odd degree points of hyperelliptic curves

We develop a reduction theory for the representation of ${\mathrm{S}\mathrm{L}}_n$ on pairs of symmetric $n\times n$ matrices. We apply this theory to the pencils of quadrics arising from divisors on hyperelliptic curves. We use these results to show that, in a density $1$ family, an odd degree point $P$ of degree at most $2g-1$ on the hyperelliptic curve $z^2=f_0{x}^{2g+2}+f_1{x}^{2g+1}y+\cdots +f_{2g+2}{y}^{2g+2}$ cannot have small Weil height.