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We prove that there is a Hermitian self-orthogonal $k$-dimensional truncatedgeneralised Reed-Solomon code of length $n \leqslant q^2$ over ${\mathbbF}_{q^2}$ if and only if there is a polynomial $g \in {\mathbb F}_{q^2}$ ofdegree at most $(q-k)q-1$ such that $g+g^q$ has $q^2-n$ distinct zeros. Thisallows us to determine the smallest $n$ for which there is a Hermitianself-orthogonal $k$-dimensional truncated generalised Reed-Solomon code oflength $n$ over ${\mathbb F}_{q^2}$, verifying a conjecture of Grassl andR\"otteler. We also provide examples of Hermitian self-orthogonal$k$-dimensional generalised Reed-Solomon codes of length $q^2+1$ over ${\mathbbF}_{q^2}$, for $k=q-1$ and $q$ an odd power of two.

Determining when a truncated generalised Reed-Solomon code is Hermitian self-orthogonal