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Abstract

If $\textbf {K}$ is a field, denote by $P(\textbf {K}, k)$ the $a\in \textbf {K}$ which are sums of $k$th powers of elements of $\textbf {K}$, by $P^{+}(\textbf {K}, k)$ the set of $a\in \textbf {K}$ which are sums of $k$th powers of totally positive elements of $\textbf {K}$. We give some simple conditions for which there exist integers $w(\textbf {K}, k)$ and $g(\textbf {K}, k)$ such that: $a \in P(\textbf {K}, k)$ implies that $a$ is the sum of at most $w(\textbf {K}, k)$ $k$th powers; $a \in P^{+}(\textbf {K}, k)$ implies that $a$ is the sum of at most $g(\textbf {K}, k)$ totally positive $k$th powers. We apply the results to characterise functions that are sums of $ k$th powers in certain function fields $\textbf {K}(X)$.