Nonnegative Polynomials and Sums of Squares
Nonnegative Polynomials and Sums of Squares
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非负多项式和平方和
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Grigoriy Blekherman
中科院分区：
文献类型：
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作者：
Grigoriy Blekherman
在最小的情况下，存在非平方和的非负多项式，我们对这种区别给出了完整的解释。平方和锥严格包含在非负多项式锥中的根本原因是d次多项式满足一定的线性关系，即Cayley-Bacharach关系，而全2d次多项式不满足这种线性关系。对于任何不是平方和的非负多项式，我们可以写出一个线性不等式来自于Cayley-Bacharach关系来证明这个事实。我们还刻画了平方和锥边界上的严格正平方和和，以及平方和锥对偶的锥的极值射线
In the smallest cases where there exist nonnegative polynomials that are not sums of squares we present a complete explanation of this distinction. The fundamental reason that the cone of sums of squares is strictly contained in the cone of nonnegative polynomials is that polynomials of degree $d$ satisfy certain linear relations, known as the Cayley-Bacharach relations, which are not satisfied by polynomials of full degree 2d. For any nonnegative polynomial that is not a sum of squares we can write down a linear inequality coming from a Cayley-Bacharach relation that certifies this fact. We also characterize strictly positive sums of squares that lie on the boundary of the cone of sums of squares and extreme rays of the cone dual to the cone of sums of squares