Waring's problem for fields

William Ellison¹

¹ 15 allée de la Borde 33450 St Sulpice et Cameyrac France

Acta Arithmetica, Tome 159 (2013) no. 4, pp. 315-330.

Résumé

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 $\textbf {K}(X)$.

Détail

DOI : 10.4064/aa159-4-2

Keywords: textbf field denote textbf textbf which sums kth powers elements textbf textbf set textbf which sums kth powers totally positive elements textbf simple conditions which there exist integers textbf textbf textbf implies sum textbf kth powers textbf implies sum textbf totally positive kth powers apply results characterise functions sums kth powers certain function fields textbf