Helton–Vinnikov determinantal representation conjecture for real zero polynomials

Let $p\in\mathbb{R}[x_1,\ldots,x_n]$ be a real zero polynomial if, for every $x\in\mathbb{R}^n$ and $\mu\in\mathbb{C}$, $p(\mu x)=0$ implies that $\mu$ is real. Suppose that $p(0)=1$. Helton–Vinnikov conjecture. There are symmetric matrices $A_1,\ldots,A_n$ such that

$$p(x_1,\ldots,x_n)=\det(I+x_1A_1+\cdots+x_nA_n).$$

The conjecture extends the Helton–Vinnikov theorem from two variables to three or more variables, allowing matrices larger than the degree. It is refuted in this paper, which constructs a family of counterexamples.