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In 1921 Siegel confirmed a conjecture of Hilbert by proving that for any number field $K$ each element of $$K_{\geq0}=\{a\in K:\ \sigma(a)\geq0\ \mbox{for all}\ \sigma\in\mathrm{Gal}(K/\mathbb Q)\}$$ can be written as $w^2+x^2+y^2+z^2$ with $w,x,y,z\in K$.

I have formulated the following conjecture which is much stronger than Siegel's result.

Conjecture 1. Let $K$ be any number field. Then each $a\in K_{\geq0}$ can be written as $w^4+x^4+y^2+z^2$ with $w,x,y,z\in K$.

This is motivated by my following conjecture.

Conjecture 2. Each $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $$w^4+\frac{x^4+y^2+z^2}{3^4}$$ with $w,x,y,z\in\mathbb N$.

For some data concerning Conjecture 2, one may visit https://oeis.org/A350860.

QUESTION. Are Conjectures 1 and 2 true? How to prove them? Can one prove Conjecture 1 for $K=\mathbb Q$?

Your comments are welcome!

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  • $\begingroup$ Written as it is, the result of Siegel is false; You need to take totally positive elements, that is, elements which are positive at every ordering of $K$, not only just one (maybe that 's what you meant , but your notation is confusiong) $\endgroup$
    – GreginGre
    Commented Jan 27, 2022 at 8:54
  • $\begingroup$ Thanks. Now we restate Conjecture 1 in a more clear form. $\endgroup$
    – Zhi-Wei Sun
    Commented Jan 27, 2022 at 9:37
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    $\begingroup$ Do you know the answer for $x^4+y^2+z^2+w^2$ in $\mathbb Q$? $\endgroup$
    – Wojowu
    Commented Jan 27, 2022 at 10:02
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    $\begingroup$ Yes, this follows from my 2017 JNT paper. $\endgroup$
    – Zhi-Wei Sun
    Commented Jan 27, 2022 at 10:16
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    $\begingroup$ In that paper I proved that any $m\in\mathbb N$ can be written as $w^4+x^2+y^2+z^2$ with $w,x,y,z\in\mathbb N$. Each $r\in\mathbb Q_{\ge0}$ can be written as $m/n=mn^3/n^4$ with $m\ge0$ and $n>0$. $\endgroup$
    – Zhi-Wei Sun
    Commented Jan 27, 2022 at 10:20

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