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This might be elementary, but I'm struggling to write the product of a sums of squares polynomial and an odd power of a variable using its Gram matrix.

Following Pablo Parrilo's notation, let $p(x,y):=[\mathbb{x}]_d^t G [\mathbb{x}]_d$ be a polynomial in $\mathbb{R}[x,y]$ which is a sum of squares of degree $2d$, where $[\mathbb{x}]_d^t:=[1,x,y,x^2,xy,y^2,x^3,x^2y,xy^2,y^3,\dots,y^d]$ is a vector of monomials and $G$ is the Gram matrix of $p(x,y)$.

How can I compute the Gram matrix of $x.p(x,y)$, for example?

I know that the matrix of the product will have dimmension $\binom{n+d+1}{d+1}$ and there will be a block of zeros somewhere, empirically.

I also know how to solve this problem for things like $x^{2s}.p(x,y)=x^s[\mathbb{x}]_d^tG[\mathbb{x}]_dx^s=[\mathbb{x}]_{d+s}^t\overline{G}[\mathbb{x}]_{d+s}$ with even degree, because then $\overline{G}$ will be sort of an extension of $G$ by adding null columns and rows conveniently.

Can I represent the matrix of $x^{2s-1}.p(x,y)$ in a similar way?

Parrilo has shown that a polynomial is SOS iff there exists a positive semidefinite Gram matrix to it, but at least does this matrix representation always exists, without assuming non-negativity?

Thanks in advance.

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