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The theorem is

A polynomial $f$ has a degree-$d$ SOS certificate if and only if there exists a positive semidefinite matrix $A$ such that for all $\in\{ 0,1 \}^$, $x\in \{ 0,1 \}^n$, $$()=⟨(1,)^{⊗/2},(1,)^{⊗/2}⟩.$$

I am confused with the tuple $(1,x)$ above, I know $⊗$ is Kronecker product but don't understand how $(1,x)$ works.

Diana
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1 Answers1

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It will make sense if $(1,x)$ is vector $(1\quad x_1 \quad x_2 \quad \dots \quad x_n)$, then the kronecker project $(1,x)^{⊗d/2}$ will be a the polynomial of $x$ with degree $d/2$. And the matrix A there gives the coefficients of each term in the polynomial.

Diana
  • 19
  • 3