论文标题
关于广场总和的程度
On the degree of varieties of sum of squares
论文作者
论文摘要
我们研究了有多少种不同的平方分解的问题，将一般的多项式$ f $与SOS级$ K $接纳。我们表明，$ f $的所有SOS分解的品种$ \ mathrm {sos} _k（f）$之间存在链接。我们利用此连接以获得$ \ mathrm {sos} _k（f）$的尺寸，并表明其度的度数是从下面的$ \ mathrm {o}（k）$限制的。特别是，对于$ k = 2 $，我们表明$ \ mathrm {sos} _2（f）$是同构与$ \ mathrm {o}（2）$的同构，因此，限制的学位成为平等。此外，我们计算SOS级$ K $多项式空间的维度，并在特殊情况下获得$ k = 2 $的学位。
We study the problem of how many different sums of squares decompositions a general polynomial $f$ with SOS-rank $k$ admits. We show that there is a link between the variety $\mathrm{SOS}_k(f)$ of all SOS-decompositions of $f$ and the orthogonal group $\mathrm{O}(k)$. We exploit this connection to obtain the dimension of $\mathrm{SOS}_k(f)$ and show that its degree is bounded from below by the degree of $\mathrm{O}(k)$. In particular, for $k=2$ we show that $\mathrm{SOS}_2(f)$ is isomorphic to $\mathrm{O}(2)$ and hence the degree bound becomes an equality. Moreover, we compute the dimension of the space of polynomials of SOS-rank $k$ and obtain the degree in the special case $k=2$.
