非負實係數多項式都能不能寫成平方和一直都被猜測，直到希爾伯特證明出任意n變數非負齊2d次多項式，都可以寫成一些d次多項式的平方和的充分必要條件為(a) n=2或(b) 2d=2又或(c) n=3且2d=4。之後，有人在射影多樣體上討論一樣的問題，發現這個實代數幾何上問題竟然和複代數幾何中一個很重要的物件極小多項式有緊密的關係。我們將用這項定理來處理一些已知的結果。

There are many guess whether a non-negative real polynomial can be written
as a sum-of-squares of polynomial. Until Hilbert proof that every non-negative
homogeneous polynomial of degree 2d in n-variables can be written as a sum-ofsquares of homogeneous polynomials of degree d if and only if either (a) n=2 or
(b) 2d=2 or (c) n=3 and 2d=4. After that, some people ask the same question
on the projective variety. They discover this problem which is the problem in the
real algebraic geometry has a closed relation with the variety of minimal degree
which is an important object in complex algebraic geometry. We will use the new
theorem to deal with some known results.

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