Let $L/\mathbb{Q}$ be a finite extension and $f\_{1},\dotsc,f\_{n}\in L[x\_{1},\dotsc,x\_{k}]$ be degree $d$ homogeneous polynomials. Is there a way to find homogenous degree $d’$ polynomials $g\_{1},\dotsc,g\_{n}\in L[x\_{1},\dotsc,x\_{k}]$ such that $f\_{1}g\_{1}+\dotsc f\_{n}g\_{n}\in \mathbb{Q}[x\_{1},\dots,x\_{k}]$ and $d’<d$?



Changing base field for sum of polynomials
Not always. Take $k=3$, $n=2$, $f\_1=\sqrt{2}x\_1^2+x\_2^2$, $f\_2=x\_3^2$.


