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Totally positive elements with m partitions exist in almost all real quadratic fields
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society , First View
- Published online by Cambridge University Press:
- 21 November 2025, pp. 1-34
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In this paper, we study partitions of totally positive integral elements
$ \alpha $ in a real quadratic field 
$ K $. We prove that for a fixed integer 
$ m \geq 1 $, an element with 
$ m $ partition exists in almost all 
$ K $. We also obtain an upper bound for the norm of 
$\alpha$ that can be represented as a sum of indecomposables in at most 
$m$ ways, completely characterize the 
$\alpha$’s represented in exactly 
$2$ ways, and subsequently apply this result to complete the search for fields containing an element with 
$ m $ partitions for 
$ 1 \leq m \leq 7 $.
