关于孙智伟一个猜想的证明

Proof of one of Sun Zhi-Wei's conjectures

本文证明了,若a、b、c和d为全实数域k中的元素且其中至少有一个是全正数,则k中每个非负元r均可表为x^(2)+y^(2)+z^(2)+w^(2)使得x、y、z和w为k中非负元并且ax+by+cz+dw为k中的平方元.若将ax+by+cz+dw为k中的平方元换为ax+by+cz+dw为k中的立方元,结论也成立.这是孙智伟的一个猜想.

We prove one of Sun Zhi-Wei's conjectures.Let k be a totally real eld and set k≥0={t∈k:t>0}.Let a,b,c,d∈k,where one of all is totally positive and n∈{2,3}.Then each r∈k≥0 can be written as x^(2)+y^(2)+z^(2)+w^(2)with x,y,z,w∈k>0 such that ax+by+cz+dw∈{t^(n):t∈k}.